Simulation of High Aspect Ratio planes in Python [SHARPy]¶

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Welcome to SHARPy (Simulation of High Aspect Ratio aeroplanes in Python)!

SHARPy is an aeroelastic analysis package currently under development at the Department of Aeronautics, Imperial College London. It can be used for the structural, aerodynamic, aeroelastic and flight dynamics analysis of flexible aircraft, flying wings and wind turbines. Amongst other capabilities, it offers the following solutions to the user:

Static aerodynamic, structural and aeroelastic solutions
Finding trim conditions for aeroelastic configurations
Nonlinear, dynamic time domain simulations under a large number of conditions such as:
- Prescribed trajectories.
- Free flight.
- Dynamic follower forces.
- Control inputs in thrust, control surface deflection…
- Arbitrary time-domain gusts, including non span-constant ones.
- Full 3D turbulent fields.
Multibody dynamics with hinges, articulations and prescribed nodal motions.
- Applicable to wind turbines.
- Hinged aircraft.
- Catapult assisted takeoffs.
Linear analysis
- Linearisation around a nonlinear equilibrium.
- Frequency response analysis.
- Asymptotic stability analysis.
Model order reduction
- Krylov-subspace reduction methods.
- Balancing reduction methods.

The modular design of SHARPy allows to simulate complex aeroelastic cases involving very flexible aircraft. The structural solver supports very complex beam arrangements, while retaining geometrical nonlinearity. The UVLM solver features different wake modelling fidelities while supporting large lifting surface deformations in a native way. Detailed information on each of the solvers is presented in their respective documentation packages.

Contents¶

SHARPy Installation Guide¶

Last revision 3 February 2020

The following step by step tutorial will guide you through the installation process of SHARPy.

Requirements¶

Operating System Requirements

SHARPy is being developed and tested on the following operating systems:

CentOS 7 and CentOS 8
Ubuntu 18.04 LTS
MacOS Mojave and Catalina

It is also available to the vast majority of operating systems that are supported by Docker, including Windows!

Required Distributions

Anaconda Python 3.7
GCC 6.0 or higher (recommended). C++ and Fortran.

Recommended Software

You may find the applications below useful, we recommend you use them but cannot provide any direct support.

HDFView to read and view .h5 files. HDF5 is the SHARPy input file format.
Paraview to visualise SHARPy’s output.

GitHub Repository

SHARPy

SHARPy can be installed from the source code available on GitHub or you can get it packed in a Docker container. If what you want is to give it a go and run some static or simple dynamic cases (and are familiar with Docker), we recommend the Docker route. If you want to check the code, modify it and compile the libraries with custom flags, build it from source (recommended).

Building SHARPy from source (release or development builds)¶

SHARPy can be built from source so that you can get the latest release or (stable) development build.

SHARPy depends on two external libraries, xbeam and UVLM. These are included as submodules to SHARPy and therefore once you initialise SHARPy you will also automatically clone the relevant versions of each library.

Set up the folder structure¶

Clone sharpy in your desired location, if you agree with the license in license.txt
```
git clone --recursive http://github.com/ImperialCollegeLondon/sharpy
```
The --recursive flag will also initialise and update the submodules SHARPy depends on, xbeam and UVLM.
We will now set up the SHARPy environment that will install other required distributions.

Setting up the Python Environment¶

SHARPy uses the Anaconda package manager to provide the necessary Python packages. These are specified in an Anaconda environment that shall be activated prior to compiling the xbeam and UVLM libraries or running any SHARPy cases.

If you do not have it, install the Anaconda Python 3 distribution
Make sure your Python version is at least 3.7:
```
python --version
```
Create the conda environment that SHARPy will use. Change environment_linux.yml to read environment_macos.yml file if you are installing SHARPy on Mac OS X
```
cd sharpy/utils
conda env create -f environment_linux.yml
cd ../..
```
We also provide a light-weight environment with the minimum required dependencies. If you’d like to use it, create the conda environment using environment_minimal.yml.
Activate the sharpy_env conda environment
```
conda activate sharpy_env
```
you need to do this before you compile the xbeam and uvlm libraries, as some dependencies are included in the conda environment.

If you would like to use the minimal environment you can run conda activate sharpy_minimal.

Quick install¶

The quick install is geared towards getting the release build of SHARPy running as quickly and simply as possible. If you would like to install a develop build or modify the compilation settings of the libraries skip to the next section.

Move into the cloned repository
```
cd sharpy
```
Ensure that the SHARPy environment is active in the session. Your terminal prompt line should begin with
```
(sharpy_env) [usr@host] $
```
If it is not the case, activate the environment. Otherwise xbeam and UVLM will not compile
```
conda activate sharpy_env
```
Create a directory build that will be used during CMake’s building process and cd into it:
```
mkdir build
cd build
```
Prepare UVLM and xbeam for compilation using gfortran and g++ in their release builds running. If you’d like to change compilers see the Custom Installation.
```
cmake ..
```
Compile the libraries
```
make install -j 4
```
where the number after the -j flag will specify how many cores to use during installation.
Finally, load the SHARPy variables
```
source bin/sharpy_vars.sh
```

You are ready to run SHARPy. Continue reading the Running SHARPy section.

Custom installation¶

These steps will show you how to compile the xbeam and UVLM libraries such that you can modify the compilation settings to your taste.

If you want to use SHARPy’s latest release, skip this step. If you would like to use the latest development work, you will need to checkout the develop branch. For more info on how we structure our development and what branches are used for what kind of features have a look at the Contributing page.
```
git checkout develop
git submodule update --recursive
```
This command will check out the develop branch and set it to track the remote origin. It will also set the submodules (xbeam and UVLM) to the right commit.
Create the conda environment that SHARPy will use. Change environment_linux.yml to read environment_macos.yml file if you are installing SHARPy on Mac OS X
```
cd sharpy/utils
conda env create -f environment_linux.yml
cd ../..
```
Activate the sharpy_env conda environment
```
conda activate sharpy_env
```
Create a directory build that will be used during CMake’s building process and cd into it:
```
mkdir build
cd build
```
Run CMake with custom flags:
1. Choose your compilers for Fortran FC and C++ CXX, for instance
```
FC=gfortran CXX=g++ cmake ..
```
  If you’d like to use the Intel compilers you can set them using:
```
FC=ifort CXX=icpc cmake ..
```
2. Alternatively, you can build the libraries in debug mode with the following flag for cmake:
```
-DCMAKE_BUILD_TYPE=Debug
```
Compile the libraries and parallelise as you prefer
```
make install -j 4
```
Finally, load the SHARPy variables
```
source bin/sharpy_vars.sh
```
This concludes the installation! Continue reading the Running SHARPy section.

Using SHARPy from a Docker container¶

Docker containers are similar to lightweight virtual machines. The SHARPy container distributed through Docker Hub is a CentOS 8 machine with the libraries compiled with gfortran and g++ and an Anaconda Python distribution.

Make sure your machine has Docker working. The instructions are here: link.

You might want to run a test in your terminal:

docker pull hello-world
docker run hello-world

If this works, you’re good to go!

First, obtain the SHARPy docker container:

docker pull fonsocarre/sharpy:stable

Now you can run it:

docker run --name sharpy -it fonsocarre/sharpy:stable

You should see a welcome dialog such as:

>>>> docker run -it fonsocarre/sharpy:stable
SHARPy added to PATH from the directory: /sharpy_dir/bin
=======================================================================
Welcome to the Docker image of SHARPy
SHARPy is located in /sharpy_dir/ and the
environment is already set up!
Copyright Imperial College London. Released under BSD 3-Clause license.
=======================================================================
SHARPy>

You are now good to go.

It is important to note that a docker container runs as an independent operating system with no access to your hard drive. If you want to copy your own files, run the container and from another terminal run:

docker cp my_file.txt sharpy:/my_file.txt     # copy from host to container
docker cp sharpy:/my_file.txt my_file.txt     # copy from container to host

The sharpy: part is the --name argument you wrote in the docker run command.

You can run the test suite once inside the container as:

cd sharpy_dir
python -m unittest

We make available two different releases: stable and experimental. The former is the latest SHARPy release. The latter is our latest development work which will include new features but with higher chances of encountering some bugs along the way. To obtain the experimental build, follow the instructions above replacing the stable tag for experimental.

Enjoy!

Running SHARPy¶

In order to run SHARPy, you need to load the conda environment and load the SHARPy variables (so your computer knows where SHARPy is). Therefore, before you run any SHARPy case:

Activate the SHARPy conda environment
```
conda activate sharpy_env
```
Load the SHARPy variables
```
source sharpy/bin/sharpy_vars.sh
```

You are now ready to run SHARPy cases from the terminal.

Automated tests¶

SHARPy uses unittests to verify the integrity of the code.

These tests can be run from the ./sharpy directory.

python -m unittest

The tests will run and you should see a success message. If you don’t… check the following options:

Check you are running the latest version. Running the following from the root directory should update to the latest release version:
- git pull
- git submodule update --init --recursive
If the tests don’t run, make sure you have followed correctly the instructions and that you managed to compile xbeam and UVLM.
If some tests fail, i.e. you get a message after the tests run saying that certain tests did not pass, please open an issue with the following information:
- Operating system
- Whether you did a Custom/quick install
- UVLM and xbeam compiler of choice
- A log of the tests that failed

The SHARPy Case Structure and input files¶

Setting up a SHARPy case

SHARPy cases are usually structured in the following way:

A generate_case.py file: contains the setup of the problem like geometry, flight conditions etc. This script creates the output files that will then be used by SHARPy, namely:
- The structural .fem.h5 file.
- The aerodynamic .aero.h5 file.
- Simulation information and settings .sharpy file.
- The dynamic forces file .dyn.h5 (when required).
- The linear input files .lininput.h5 (when required).
- The ROM settings file .rom.h5 (when required).
See the chapter on the case files for a detailed description on the contents of each one.Data is exchanged in binary format by means of .h5 files that make the transmission efficient between the different languages of the required libraries. To view these .h5 files, a viewer like HDF5 is recommended.
The h5 files contain data of the FEM, aerodynamics, dynamic conditions. They are later read by SHARPy.
The .sharpy file contains the settings for SHARPy and is the file that is parsed to SHARPy.

To run a SHARPy case

SHARPy cases are therefore usually ran in the following way:

Create a generate_case.py file following the provided templates.
Run it to produce the .h5 files and the .sharpy files.
```
(sharpy_env) python generate_case.py
```
Run SHARPy (ensure the environment is activated).
```
(sharpy_env) sharpy case.sharpy
```

Output¶

By default, the output is located in the output folder.

The contents of the folder will typically be a beam and aero folders, which contain the output data that can then be loaded in Paraview. These are the .vtu format files that can be used with Paraview.

Running (and modifiying) a test case¶

This command generates the required files for running a static, clamped beam case that is used as part of code verification:
```
cd ../sharpy
python ./tests/xbeam/geradin/generate_geradin.py
```

Now you should see a success message, and if you check the ./tests/xbeam/geradin/ folder, you should see two new files:

geradin_cardona.sharpy
geradin_cardona.fem.h5

Try to open the sharpy file with a plain text editor and have a quick look. The sharpy file is the main settings file. We’ll get deeper into this later.

If you try to open the fem.h5 file, you’ll get an error or something meaningless. This is because the structural data is stored in HDF5 format, which is compressed binary.

Run it (part 1)

The sharpy call is:

# Make sure that the sharpy_env conda environment is active
sharpy <path to solver file>

Results (part 1)

Since this is a test case, there is no output directly to screen.

We will therefore change this setting first. In the generate_geradin.py file, look for the SHARPy setting write_screen and set it to on. This will output the progress of the execution to the terminal.

We would also like to create a Paraview file to view the beam deformation. Append the post-processor BeamPlot to the end of the SHARPy setting flow, which is a list. This will run the post-processor and plot the beam in Paraview format with the settings specified in the generate_geradin.py file under config['BeamPlot].
Run (part 2)

Now that we have made these modifications, run again the generation script:
```
python ./tests/xbeam/geradin/generate_geradin.py
```
Check the solver file geradin.sharpy and look for the settings we just changed. Make sure they read what we wanted.

You are now ready to run the case again:
```
# Make sure that the sharpy_env conda environment is active
sharpy <path to solver file>
```
Post-processing

After a successful execution, you should a long display of information in the terminal as the case is being executed.

The deformed beam will have been written in a .vtu file and will be located in the output/ folder (or where you specified in the settings) which you can open using Paraview.

In the output directory you will also note a folder named WriteVariablesTime which outputs certain variables as a function of time to a .dat file. In this case, the beam tip position deflection and rotation is written. Check the values of those files and look for the following result:
```
    Pos_def:
	      4.403530 0.000000 -2.159692
    Psi_def:
	      0.000000 0.672006 0.000000
```
FYI, the correct solution for this test case by Geradin and Cardona is Delta R_3 = -2.159 m and Psi_2 = 0.6720 rad.

Congratulations, you’ve run your first case. You can now check the Examples section for further cases.

Capabilities¶

This is just the tip of the iceberg, possibilities are nearly endless and once you understand how SHARPy’s modular interface works, you will be capable of running very complex simulations.

Very flexible aircraft nonlinear aeroelasticity¶

The modular design of SHARPy allows to simulate complex aeroelastic cases involving very flexible aircraft. The structural solver supports very complex beam arrangements, while retaining geometrical nonlinearity. The UVLM solver features different wake modelling fidelities while supporting large lifting surface deformations in a native way.

Among the problems studied, a few interesting ones, in no particular order are:

Catapult take off of a very flexible aircraft analysis [Paper]. In this type of simulations, a PID controller was used in order to enforce displacements and velocities in a number of structural nodes (the clamping points). Then, several take off strategies were studied in order to analyse the influence of the structural stiffness in this kind of procedures. This case is a very good example of the type of problems where nonlinear aeroelasticity is essential.

_images/hale_cruise.png Catapult Takeoff of Flexible Aircraft

Flight in full 3D atmospheric boundary layer (to be published). A very flexible aircraft is flown immersed in a turbulent boundary layer obtained from HPC LES simulations. The results are compared against simpler turbulence models such as von Karman and Kaimal. Intermittency and coherence features in the LES field are absent or less remarkable in the synthetic turbulence fields.

_images/hale_turb.jpeg HALE Aircraft in a Turbulent Field

Lateral gust reponse of a realistic very flexible aircraft. For this problem (to be published), a realistic very flexible aircraft (University of Michigan X-HALE) model has been created in SHARPy and validated against their own aeroelastic solver for static and dynamic cases. A set of vertical and lateral gust responses have been simulated.

_images/xhale.png X-HALE

Wind turbine aeroelasticity¶

SHARPy is suitable to simulate wind turbine aeroelasticity.

On the structural side, it accounts for material anisotropy which is needed to characterize composite blades and for geometrically non-linear deformations observed in current blades due to the increasing length and flexibility. Both rigid and flexible simulations can be performed and the structural modes can be computed accounting for rotational effects (Campbell diagrams). The rotor-tower interaction is modelled through a multibody approach based on the theory of Lagrange multipliers. Finally, the tower base can be fixed or subjected to prescribed linear and angular velocities.

On the aerodynamic side, the use of potential flow theory allows the characterization of flow unsteadiness at a reasonable computational cost. Specifically, steady and dynamic simulations can be performed. The steady simulations are carried out in a non-inertial frame of reference linked to the rotor under uniform steady wind with the assumption of prescribed helicoidal wake. On the other hand, dynamic simulations can be enriched with a wide variety of incoming winds such as shear and yaw. Moreover, the wake shape can be freely computed under no assumptions accounting for self-induction and wake expansion or can be prescribed to an helicoidal shape for computational efficiency.

_images/turbine.png Wind Turbine

Model Order Reduction¶

Numerical models of physical phenomena require fine discretisations to show convergence and agreement with their real counterparts, and, in the case of SHARPy’s aeroelastic systems, hundreds of thousands of states are not an uncommon encounter. However, modern hardware or the use of these models for other applications such as controller synthesis may limit their size, and we must turn to model order reduction techniques to achieve lower dimensional representations that can then be used.

SHARPy offers several model order reduction methods to reduce the initially large system to a lower dimension, attending to the user’s requirements of numerical efficiency or global error bound.

Krylov Methods for Model Order Reduction - Moment Matching¶

Model reduction by moment matching can be seen as approximating a transfer function through a power series expansion about a user defined point in the complex plane. The reduction by projection retains the moments between the full and reduced systems as long as the projection matrices span certain Krylov subspaces dependant on the expansion point and the system’s matrices. This can be taken advantage of, in particular for aeroelastic applications where the interest resides in the low frequency behaviour of the system, the ROM can be expanded about these low frequency points discarding accuracy higher up the frequency spectrum.

Example 1 - Aerodynamics - Frequency response of a high AR flat plate subject to a sinusoidal gust¶

The objective is to compare SHARPy’s solution of a very high aspect ratio flat plate subject to a sinusoidal gust to the closed form solution obtained by Sears (1944 - Ref). SHARPy’s inherent 3D nature makes comparing results to the 2D solution require very high aspect ratio wings with fine discretisations, resulting in very large state space models. In this case, we would like to utilise a Krylov ROM to approximate the low frequency behaviour and perform a frequency response analysis on the reduced system, since it would represent too much computational cost if it were performed on the full system.

The full order model was reduced utilising Krylov methods, in particular the Arnoldi iteration, with an expansion about zero frequency to produce the following result.

_images/sears.png Sears Gust Bode Plot

As it can be seen from the image above, the ROM approximates well the low frequency, quasi-steady state and loses accuracy as the frequency is increased, just as intended. Still, perfect matching is never achieved even at the expansion frequency given the 3D nature of the wing compared to the 2D analytical solution.

Publications¶

SHARPy has been used in many technical papers that have been both published in Journals and presented at conferences. Here we present a list of past papers which have used SHARPy for research purposes:

2022¶

Goizueta, N., Wynn, A., Palacios, R., Drachinsky, A., Raveh, D. E. (2022). Flutter Predictions for Very Flexible Wing Wind Tunnel Test. Journal of Aircraft. Article in Advance. https://doi.org/10.2514/1.C036710
Muñoz-Simón A, Wynn A, Palacios R (2022). Some modelling improvements for prediction of wind turbine rotor loads in turbulent wind. Wind Energy: 25(2), pp. 333-353. https://doi.org/10.1002/we.2675
Goizueta, N., Wynn, A., Palacios, R. (2022). Fast flutter evaluation of very flexible wing using interpolation on an optimal training dataset. AIAA SciTech Forum. https://doi.org/10.2514/6.2022-1345
Düssler, S., Goizueta, N., Muñoz-Simón, A., & Palacios, R. (2022). Modelling and Numerical Enhancements on a UVLM for Nonlinear Aeroelastic Simulation. AIAA SciTech Forum. https://doi.org/10.2514/6.2022-2455
Cea, A., Palacios, R. (2022). Parametric Reduced Order Models for Aeroelastic Design of Very Flexible Aircraft. AIAA SciTech Forum. https://doi.org/10.2514/6.2022-0727
Wynn, A., Artola, M., Palacios, R. (2022). Nonlinear optimal control for gust load alleviation with a physics-constrained data-driven internal model. AIAA SciTech Forum. https://doi.org/10.2514/6.2022-0442

2021¶

Artola, M., Goizueta, N., Wynn, A., & Palacios, R. (2021). Aeroelastic Control and Estimation with a Minimal Nonlinear Modal Description. AIAA Journal: 59(7), pp. 2697–2713. https://doi.org/10.2514/1.j060018
Artola, M., Goizueta, N., Wynn, A., & Palacios, R. (2021). Proof of Concept for a Hardware-in-the-Loop Nonlinear. In AIAA SciTech Forum (pp. 1–26). https://doi.org/10.2514/6.2021-1392
Goizueta, N., Drachinsky, A., Wynn, A., Raveh, D. E., & Palacios, R. (2021). Flutter prediction for a very flexible wing wind tunnel test. In AIAA SciTech Forum (pp. 1–17). https://doi.org/10.2514/6.2021-1711
Goizueta, N., Wynn, A., & Palacios, R. (2021). Parametric Krylov-based order reduction of aircraft aeroelastic models. In AIAA SciTech Forum (pp. 1–25). https://doi.org/10.2514/6.2021-1798

2020¶

Muñoz-Simón, A., Palacios, R., & Wynn, A. (2020). Benchmarking different fidelities in wind turbine aerodynamics under yaw. Journal of Physics: Conference Series, 1618, 42017. https://doi.org/10.1088/1742-6596/1618/4/042017
del Carre, A., & Palacios, R. (2020). Simulation and Optimization of Takeoff Maneuvers of Very Flexible Aircraft. Journal of Aircraft: 57(6) 1097-1110. https://doi.org/10.2514/1.C035901
Maraniello, S. & Palacios, R. (2020). Parametric Reduced-Order Modeling of the Unsteady Vortex-Lattice Method. AIAA Journal, 58(5), 2206-2220. https://doi.org/10.2514/1.J058894
Deskos, G., del Carre, A., & Palacios, R. (2020). Assessment of low-altitude atmospheric turbulence models for aircraft aeroelasticity. Journal of Fluids and Structures, 95, 102981. https://doi.org/10.1016/j.jfluidstructs.2020.102981
Goizueta, Norberto, del Carre, Alfonso, Muñoz-Simón, Arturo, & Palacios, Rafael. (2020, February). SHARPy: from a research code to an open-source software tool for the simulation of very flexible aircraft. RSLondonSouthEast 2020 Conference. Zenodo: http://doi.org/10.5281/zenodo.3641216
Del Carre, A., Deskos, G., & Palacios, R. (2020). Realistic turbulence effects in low altitude dynamics of very flexible aircraft. In AIAA SciTech Forum (pp. 1–18). https://doi.org/10.2514/6.2020-1187
Artola, M., Goizueta, N., Wynn, A., & Palacios, R. (2020). Modal-Based Nonlinear Estimation and Control for Highly Flexible Aeroelastic Systems. In AIAA SciTech Forum (pp. 1–23). https://doi.org/10.2514/6.2020-1192
Muñoz-Simón, A., Wynn, A., & Palacios, R. (2020). Unsteady and three-dimensional aerodynamic effects on wind turbine rotor loads. In AIAA SciTech Forum. https://doi.org/10.2514/6.2020-0991

2019¶

del Carre, A., Muñoz-Simón, A., Goizueta, N., & Palacios, R. (2019). SHARPy : A dynamic aeroelastic simulation toolbox for very flexible aircraft and wind turbines. Journal of Open Source Software, 4(44), 1885. https://doi.org/10.21105/joss.01885
del Carre, A., Teixeira, P. C., Palacios, R., & Cesnik, C. E. S. (2019). Nonlinear Response of a Very Flexible Aircraft Under Lateral Gust. In International Forum on Aeroelasticity and Structural Dynamics.
del Carre, A., & Palacios, R. (2019). Efficient Time-Domain Simulations in Nonlinear Aeroelasticity. In AIAA Scitech Forum (pp. 1–20). https://doi.org/10.2514/6.2019-2038
Maraniello, S., & Palacios, R. (2019). State-Space Realizations and Internal Balancing in Potential-Flow Aerodynamics with Arbitrary Kinematics. AIAA Journal, 57(6), 2308-2321. https://doi.org/10.2514/1.J058153

Examples¶

A set of SHARPy examples created with Jupyter Notebooks is provided for users to interact and modify cases running on SHARPy.

Flutter Analysis of a Goland Wing using the SHARPy Linear Solver¶

This is an example using SHARPy to find the flutter speed of a Goland wing by:

Calculating aerodynamic forces and deflections using a nonlinear solver
Linearising about this reference condition
Creating a reduced order model of the linearised aerodynamics
Evaluate the stability of the linearised aeroelastic system at different velocities

References¶

Maraniello, S., & Palacios, R. (2019). State-Space Realizations and Internal Balancing in Potential-Flow Aerodynamics with Arbitrary Kinematics. AIAA Journal, 57(6), 1–14. https://doi.org/10.2514/1.J058153

Required Packages¶

[1]:

import numpy as np
import matplotlib.pyplot as plt
import os
import sys
import cases.templates.flying_wings as wings  # See this package for the Goland wing structural and aerodynamic definition
import sharpy.sharpy_main  # used to run SHARPy from Jupyter

Problem Set-up¶

Velocity¶

The UVLM is assembled in normalised time at a velocity of $1 m/s$. The only matrices that need updating then with free stream velocity are the structural matrices, which is significantly cheaper to do than to update the UVLM.

[2]:

u_inf = 1.
alpha_deg = 0.
rho = 1.02
num_modes = 4

Discretisation¶

Note: To achieve convergence of the flutter results with the ones found in the literature, a significant discretisation may be required. If you are running this notebook for the first time, set M = 4 initially to verify that your system can perform!

[3]:

M = 16
N = 32
M_star_fact = 10

ROM¶

A moment-matching (Krylov subspace) model order reduction technique is employed. This ROM method offers the ability to interpolate the transfer functions at a desired point in the complex plane. See the ROM documentation pages for more info.

Note: this ROM method matches the transfer function but does not guarantee stability. Therefore the resulting system may be unstable. These unstable modes may appear far in the right hand plane but will not affect the flutter speed calculations.

[4]:

c_ref = 1.8288 # Goland wing reference chord. Used for frequency normalisation
rom_settings = dict()
rom_settings['algorithm'] = 'mimo_rational_arnoldi'  # reduction algorithm
rom_settings['r'] = 6  # Krylov subspace order
frequency_continuous_k = np.array([0.])  # Interpolation point in the complex plane with reduced frequency units
frequency_continuous_w = 2 * u_inf * frequency_continuous_k / c_ref
rom_settings['frequency'] = frequency_continuous_w

Case Admin¶

[5]:

case_name = 'goland_cs'
case_nlin_info = 'M%dN%dMs%d_nmodes%d' % (M, N, M_star_fact, num_modes)
case_rom_info = 'rom_MIMORA_r%d_sig%04d_%04dj' % (rom_settings['r'], frequency_continuous_k[-1].real * 100,
                                                  frequency_continuous_k[-1].imag * 100)

case_name += case_nlin_info + case_rom_info

route_test_dir = os.path.abspath('')

print('The case to run will be: %s' % case_name)
print('Case files will be saved in ./cases/%s' %case_name)
print('Output files will be saved in ./output/%s/' %case_name)

The case to run will be: goland_csM16N32Ms10_nmodes4rom_MIMORA_r6_sig0000_0000j
Case files will be saved in ./cases/goland_csM16N32Ms10_nmodes4rom_MIMORA_r6_sig0000_0000j
Output files will be saved in ./output/goland_csM16N32Ms10_nmodes4rom_MIMORA_r6_sig0000_0000j/

Simulation Set-Up¶

Goland Wing¶

ws is an instance of a Goland wing with a control surface. Reference the template file cases.templates.flying_wings.GolandControlSurface for more info on the geometrical, structural and aerodynamic definition of the Goland wing here used.

[6]:

ws = wings.GolandControlSurface(M=M,
                                N=N,
                                Mstar_fact=M_star_fact,
                                u_inf=u_inf,
                                alpha=alpha_deg,
                                cs_deflection=[0, 0],
                                rho=rho,
                                sweep=0,
                                physical_time=2,
                                n_surfaces=2,
                                route=route_test_dir + '/cases',
                                case_name=case_name)

ws.clean_test_files()
ws.update_derived_params()
ws.set_default_config_dict()

ws.generate_aero_file()
ws.generate_fem_file()

Simulation Settings¶

The settings for each of the solvers are now set. For a detailed description on them please reference their respective documentation pages

SHARPy Settings¶

The most important setting is the flow list. It tells SHARPy which solvers to run and in which order.

[7]:

ws.config['SHARPy'] = {
    'flow':
        ['BeamLoader', 'AerogridLoader',
         'StaticCoupled',
         'AerogridPlot',
         'BeamPlot',
         'Modal',
         'LinearAssembler',
         'AsymptoticStability',
         ],
    'case': ws.case_name, 'route': ws.route,
    'write_screen': 'on', 'write_log': 'on',
    'log_folder': route_test_dir + '/output/',
    'log_file': ws.case_name + '.log'}

Beam Loader Settings¶

[8]:

ws.config['BeamLoader'] = {
    'unsteady': 'off',
    'orientation': ws.quat}

Aerogrid Loader Settings¶

[9]:

ws.config['AerogridLoader'] = {
    'unsteady': 'off',
    'aligned_grid': 'on',
    'mstar': ws.Mstar_fact * ws.M,
    'freestream_dir': ws.u_inf_direction,
    'wake_shape_generator': 'StraightWake',
    'wake_shape_generator_input': {'u_inf': ws.u_inf,
                                    'u_inf_direction': ws.u_inf_direction,
                                    'dt': ws.dt}}

Static Coupled Solver¶

[10]:

ws.config['StaticCoupled'] = {
    'print_info': 'on',
    'max_iter': 200,
    'n_load_steps': 1,
    'tolerance': 1e-10,
    'relaxation_factor': 0.,
    'aero_solver': 'StaticUvlm',
    'aero_solver_settings': {
        'rho': ws.rho,
        'print_info': 'off',
        'horseshoe': 'off',
        'num_cores': 4,
        'n_rollup': 0,
        'rollup_dt': ws.dt,
        'rollup_aic_refresh': 1,
        'rollup_tolerance': 1e-4,
        'velocity_field_generator': 'SteadyVelocityField',
        'velocity_field_input': {
            'u_inf': ws.u_inf,
            'u_inf_direction': ws.u_inf_direction}},
    'structural_solver': 'NonLinearStatic',
    'structural_solver_settings': {'print_info': 'off',
                                   'max_iterations': 150,
                                   'num_load_steps': 4,
                                   'delta_curved': 1e-1,
                                   'min_delta': 1e-10,
                                   'gravity_on': 'on',
                                   'gravity': 9.81}}

AerogridPlot Settings¶

[11]:

ws.config['AerogridPlot'] = {'include_rbm': 'off',
                             'include_applied_forces': 'on',
                             'minus_m_star': 0}

BeamPlot Settings¶

[12]:

ws.config['BeamPlot'] = {'include_rbm': 'off',
                         'include_applied_forces': 'on'}

Linear System Assembly Settings¶

[14]:

ws.config['LinearAssembler'] = {'linear_system': 'LinearAeroelastic',
                                'linear_system_settings': {
                                    'beam_settings': {'modal_projection': 'on',
                                                      'inout_coords': 'modes',
                                                      'discrete_time': 'on',
                                                      'newmark_damp': 0.5e-4,
                                                      'discr_method': 'newmark',
                                                      'dt': ws.dt,
                                                      'proj_modes': 'undamped',
                                                      'use_euler': 'off',
                                                      'num_modes': num_modes,
                                                      'print_info': 'on',
                                                      'gravity': 'on',
                                                      'remove_sym_modes': 'on',
                                                      'remove_dofs': []},
                                    'aero_settings': {'dt': ws.dt,
                                                      'ScalingDict': {'length': 0.5 * ws.c_ref,
                                                                      'speed': u_inf,
                                                                      'density': rho},
                                                      'integr_order': 2,
                                                      'density': ws.rho,
                                                      'remove_predictor': 'on',
                                                      'use_sparse': 'on',
                                                      'remove_inputs': ['u_gust'],
                                                      'rom_method': ['Krylov'],
                                                      'rom_method_settings': {'Krylov': rom_settings}},
                                    }}

Asymptotic Stability Analysis Settings¶

[15]:

ws.config['AsymptoticStability'] = {'print_info': True,
                                    'velocity_analysis': [100, 180, 81],
                                   'modes_to_plot': []}

[16]:

ws.config.write()

Run SHARPy¶

[17]:

sharpy.sharpy_main.main(['', ws.route + ws.case_name + '.sharpy'])

--------------------------------------------------------------------------------
            ######  ##     ##    ###    ########  ########  ##    ##
           ##    ## ##     ##   ## ##   ##     ## ##     ##  ##  ##
           ##       ##     ##  ##   ##  ##     ## ##     ##   ####
            ######  ######### ##     ## ########  ########     ##
                 ## ##     ## ######### ##   ##   ##           ##
           ##    ## ##     ## ##     ## ##    ##  ##           ##
            ######  ##     ## ##     ## ##     ## ##           ##
--------------------------------------------------------------------------------
Aeroelastics Lab, Aeronautics Department.
    Copyright (c), Imperial College London.
    All rights reserved.
    License available at https://github.com/imperialcollegelondon/sharpy
Running SHARPy from /home/ng213/2TB/pazy_code/pazy-sharpy/lib/sharpy/docs/source/content/example_notebooks
SHARPy being run is in /home/ng213/2TB/pazy_code/pazy-sharpy/lib/sharpy
The branch being run is dev_setting_error
The version and commit hash are: v1.2.1-339-g156c731-156c731
SHARPy output folder set
     /home/ng213/2TB/pazy_code/pazy-sharpy/lib/sharpy/docs/source/content/example_notebooks/output//goland_csM16N32Ms10_nmodes4rom_MIMORA_r6_sig0000_0000j/
Generating an instance of BeamLoader
Variable for_pos has no assigned value in the settings file.
    will default to the value: [0.0, 0, 0]
Generating an instance of AerogridLoader
Variable control_surface_deflection has no assigned value in the settings file.
    will default to the value: []
Variable control_surface_deflection_generator_settings has no assigned value in the settings file.
    will default to the value: {}
Variable dx1 has no assigned value in the settings file.
    will default to the value: -1.0
Variable ndx1 has no assigned value in the settings file.
    will default to the value: 1
Variable r has no assigned value in the settings file.
    will default to the value: 1.0
Variable dxmax has no assigned value in the settings file.
    will default to the value: -1.0
The aerodynamic grid contains 2 surfaces
  Surface 0, M=16, N=16
     Wake 0, M=160, N=16
  Surface 1, M=16, N=16
     Wake 1, M=160, N=16
  In total: 512 bound panels
  In total: 5120 wake panels
  Total number of panels = 5632
Generating an instance of StaticCoupled
Variable correct_forces_method has no assigned value in the settings file.
    will default to the value: 
Variable runtime_generators has no assigned value in the settings file.
    will default to the value: {}
Generating an instance of NonLinearStatic
Variable newmark_damp has no assigned value in the settings file.
    will default to the value: 0.0001
Variable gravity_dir has no assigned value in the settings file.
    will default to the value: [0.0, 0.0, 1.0]
Variable relaxation_factor has no assigned value in the settings file.
    will default to the value: 0.3
Variable dt has no assigned value in the settings file.
    will default to the value: 0.01
Variable num_steps has no assigned value in the settings file.
    will default to the value: 500
Variable initial_position has no assigned value in the settings file.
    will default to the value: [0. 0. 0.]
Generating an instance of StaticUvlm
Variable iterative_solver has no assigned value in the settings file.
    will default to the value: False
Variable iterative_tol has no assigned value in the settings file.
    will default to the value: 0.0001
Variable iterative_precond has no assigned value in the settings file.
    will default to the value: False
Variable cfl1 has no assigned value in the settings file.
    will default to the value: True
Variable vortex_radius has no assigned value in the settings file.
    will default to the value: 1e-06
Variable vortex_radius_wake_ind has no assigned value in the settings file.
    will default to the value: 1e-06
Variable rbm_vel_g has no assigned value in the settings file.
    will default to the value: [0.0, 0.0, 0.0, 0.0, 0.0, 0.0]
Variable centre_rot_g has no assigned value in the settings file.
    will default to the value: [0.0, 0.0, 0.0]



|=====|=====|============|==========|==========|==========|==========|==========|==========|
|iter |step | log10(res) |    Fx    |    Fy    |    Fz    |    Mx    |    My    |    Mz    |
|=====|=====|============|==========|==========|==========|==========|==========|==========|
|  0  |  0  |  0.00000   |  0.0000  | -0.0000  |-4271.0417|  0.0000  | 781.0842 |  0.0000  |
|  1  |  0  | -11.89144  |  0.0000  | -0.0000  |-4271.0039|  0.0000  | 781.0906 |  0.0000  |
Generating an instance of AerogridPlot
Variable include_forward_motion has no assigned value in the settings file.
    will default to the value: False
Variable include_unsteady_applied_forces has no assigned value in the settings file.
    will default to the value: False
Variable name_prefix has no assigned value in the settings file.
    will default to the value: 
Variable u_inf has no assigned value in the settings file.
    will default to the value: 0.0
Variable dt has no assigned value in the settings file.
    will default to the value: 0.0
Variable include_velocities has no assigned value in the settings file.
    will default to the value: False
Variable include_incidence_angle has no assigned value in the settings file.
    will default to the value: False
Variable num_cores has no assigned value in the settings file.
    will default to the value: 1
Variable vortex_radius has no assigned value in the settings file.
    will default to the value: 1e-06
...Finished
Generating an instance of BeamPlot
Variable include_FoR has no assigned value in the settings file.
    will default to the value: False
Variable include_applied_moments has no assigned value in the settings file.
    will default to the value: True
Variable name_prefix has no assigned value in the settings file.
    will default to the value: 
Variable output_rbm has no assigned value in the settings file.
    will default to the value: True
...Finished
Generating an instance of Modal
Variable print_info has no assigned value in the settings file.
    will default to the value: True
Variable delta_curved has no assigned value in the settings file.
    will default to the value: 0.01
Variable use_custom_timestep has no assigned value in the settings file.
    will default to the value: -1
Structural eigenvalues



|==============|==============|==============|==============|==============|==============|==============|
|     mode     |  eval_real   |  eval_imag   | freq_n (Hz)  | freq_d (Hz)  |   damping    |  period (s)  |
|==============|==============|==============|==============|==============|==============|==============|
|      0       |   0.000000   |  48.067396   |   7.650164   |   7.650164   |  -0.000000   |   0.130716   |
|      1       |   0.000000   |  48.067398   |   7.650164   |   7.650164   |  -0.000000   |   0.130716   |
|      2       |   0.000000   |  95.685736   |  15.228858   |  15.228858   |  -0.000000   |   0.065665   |
|      3       |   0.000000   |  95.685754   |  15.228861   |  15.228861   |  -0.000000   |   0.065665   |
|      4       |   0.000000   |  243.144471  |  38.697644   |  38.697644   |  -0.000000   |   0.025841   |
|      5       |   0.000000   |  243.144477  |  38.697645   |  38.697645   |  -0.000000   |   0.025841   |
|      6       |   0.000000   |  343.801136  |  54.717650   |  54.717650   |  -0.000000   |   0.018276   |
|      7       |   0.000000   |  343.801137  |  54.717650   |  54.717650   |  -0.000000   |   0.018276   |
|      8       |   0.000000   |  443.324608  |  70.557303   |  70.557303   |  -0.000000   |   0.014173   |
|      9       |   0.000000   |  443.324619  |  70.557304   |  70.557304   |  -0.000000   |   0.014173   |
|      10      |   0.000000   |  461.992869  |  73.528449   |  73.528449   |  -0.000000   |   0.013600   |
|      11      |   0.000000   |  461.992869  |  73.528449   |  73.528449   |  -0.000000   |   0.013600   |
|      12      |   0.000000   |  601.126871  |  95.672313   |  95.672313   |  -0.000000   |   0.010452   |
|      13      |   0.000000   |  601.126873  |  95.672313   |  95.672313   |  -0.000000   |   0.010452   |
|      14      |   0.000000   |  782.997645  |  124.617946  |  124.617946  |  -0.000000   |   0.008025   |
|      15      |   0.000000   |  782.997649  |  124.617946  |  124.617946  |  -0.000000   |   0.008025   |
|      16      |   0.000000   |  917.191257  |  145.975522  |  145.975522  |  -0.000000   |   0.006850   |
|      17      |   0.000000   |  917.191259  |  145.975523  |  145.975523  |  -0.000000   |   0.006850   |
|      18      |   0.000000   |  975.005694  |  155.176976  |  155.176976  |  -0.000000   |   0.006444   |
|      19      |   0.000000   |  975.005699  |  155.176977  |  155.176977  |  -0.000000   |   0.006444   |
Generating an instance of LinearAssembler
Variable linearisation_tstep has no assigned value in the settings file.
    will default to the value: -1
Variable modal_tstep has no assigned value in the settings file.
    will default to the value: -1
Variable inout_coordinates has no assigned value in the settings file.
    will default to the value: 
Variable retain_inputs has no assigned value in the settings file.
    will default to the value: []
Variable retain_outputs has no assigned value in the settings file.
    will default to the value: []
Generating an instance of LinearAeroelastic
Variable uvlm_filename has no assigned value in the settings file.
    will default to the value: 
Variable track_body has no assigned value in the settings file.
    will default to the value: True
Variable use_euler has no assigned value in the settings file.
    will default to the value: False
Generating an instance of LinearUVLM
Variable gust_assembler has no assigned value in the settings file.
    will default to the value: 
Variable vortex_radius has no assigned value in the settings file.
    will default to the value: 1e-06
Variable cfl1 has no assigned value in the settings file.
    will default to the value: True
Variable velocity_field_generator has no assigned value in the settings file.
    will default to the value: SteadyVelocityField
Variable velocity_field_input has no assigned value in the settings file.
    will default to the value: {}
Variable physical_model has no assigned value in the settings file.
    will default to the value: True
Variable track_body has no assigned value in the settings file.
    will default to the value: False
Variable track_body_number has no assigned value in the settings file.
    will default to the value: -1
Initialising Static linear UVLM solver class...
                        ...done in 1.35 sec
Generating an instance of Krylov
Variable print_info has no assigned value in the settings file.
    will default to the value: True
Variable single_side has no assigned value in the settings file.
    will default to the value: 
Variable tangent_input_file has no assigned value in the settings file.
    will default to the value: 
Variable restart_arnoldi has no assigned value in the settings file.
    will default to the value: False
Initialising Krylov Model Order Reduction
State-space realisation of UVLM equations started...
Computing wake propagation matrix with CFL1=True

/home/ng213/anaconda3/envs/sharpy_env/lib/python3.7/site-packages/scipy/sparse/_index.py:126: SparseEfficiencyWarning: Changing the sparsity structure of a csc_matrix is expensive. lil_matrix is more efficient.
  self._set_arrayXarray(i, j, x)

     state-space model produced in form:
                     h_{n+1} = A h_{n} + B u_{n}
                     with:
     x_n = h_n + Bp u_n
                        ...done in 17.25 sec
Scaling UVLM system with reference time 0.914400s
Non-dimensional time step set (0.125000)
System scaled in 31.698308s
Generating an instance of LinearBeam
Warning, projecting system with damping onto undamped modes

Linearising gravity terms...
     M = 7.26 kg
     X_CG A -> 0.00 0.00 -0.00
Node  1      -> B -0.000 -0.116 0.000
                     -> A 0.116 0.381 -0.000
                     -> G 0.116 0.381 -0.000
     Node mass:
             Matrix: 14.5125
Node  2      -> B -0.000 -0.116 0.000
                     -> A 0.116 0.762 -0.000
                     -> G 0.116 0.762 -0.000
     Node mass:
             Matrix: 7.2563
Node  3      -> B -0.000 -0.116 0.000
                     -> A 0.116 1.143 -0.000
                     -> G 0.116 1.143 -0.000
     Node mass:
             Matrix: 14.5125
Node  4      -> B -0.000 -0.116 0.000
                     -> A 0.116 1.524 -0.001
                     -> G 0.116 1.524 -0.001
     Node mass:
             Matrix: 7.2563
Node  5      -> B -0.000 -0.116 0.000
                     -> A 0.116 1.905 -0.001
                     -> G 0.116 1.905 -0.001
     Node mass:
             Matrix: 14.5125
Node  6      -> B -0.000 -0.116 0.000
                     -> A 0.116 2.286 -0.001
                     -> G 0.116 2.286 -0.001
     Node mass:
             Matrix: 7.2563
Node  7      -> B -0.000 -0.116 0.000
                     -> A 0.116 2.667 -0.002
                     -> G 0.116 2.667 -0.002
     Node mass:
             Matrix: 14.5125
Node  8      -> B -0.000 -0.116 0.000
                     -> A 0.116 3.048 -0.002
                     -> G 0.116 3.048 -0.002
     Node mass:
             Matrix: 7.2563
Node  9      -> B -0.000 -0.116 0.000
                     -> A 0.116 3.429 -0.003
                     -> G 0.116 3.429 -0.003
     Node mass:
             Matrix: 14.5125
Node 10      -> B -0.000 -0.116 0.000
                     -> A 0.116 3.810 -0.003
                     -> G 0.116 3.810 -0.003
     Node mass:
             Matrix: 7.2563
Node 11      -> B -0.000 -0.116 0.000
                     -> A 0.116 4.191 -0.004
                     -> G 0.116 4.191 -0.004
     Node mass:
             Matrix: 14.5125
Node 12      -> B -0.000 -0.116 0.000
                     -> A 0.116 4.572 -0.004
                     -> G 0.116 4.572 -0.004
     Node mass:
             Matrix: 7.2563
Node 13      -> B -0.000 -0.116 0.000
                     -> A 0.116 4.953 -0.005
                     -> G 0.116 4.953 -0.005
     Node mass:
             Matrix: 14.5125
Node 14      -> B -0.000 -0.116 0.000
                     -> A 0.116 5.334 -0.005
                     -> G 0.116 5.334 -0.005
     Node mass:
             Matrix: 7.2563
Node 15      -> B -0.000 -0.116 0.000
                     -> A 0.116 5.715 -0.006
                     -> G 0.116 5.715 -0.006
     Node mass:
             Matrix: 14.5125
Node 16      -> B -0.000 -0.116 0.000
                     -> A 0.116 6.096 -0.006
                     -> G 0.116 6.096 -0.006
     Node mass:
             Matrix: 3.6281
Node 17      -> B -0.000 -0.116 -0.000
                     -> A 0.116 -6.096 -0.006
                     -> G 0.116 -6.096 -0.006
     Node mass:
             Matrix: 3.6281
Node 18      -> B -0.000 -0.116 -0.000
                     -> A 0.116 -5.715 -0.006
                     -> G 0.116 -5.715 -0.006
     Node mass:
             Matrix: 14.5125
Node 19      -> B -0.000 -0.116 -0.000
                     -> A 0.116 -5.334 -0.005
                     -> G 0.116 -5.334 -0.005
     Node mass:
             Matrix: 7.2563
Node 20      -> B -0.000 -0.116 -0.000
                     -> A 0.116 -4.953 -0.005
                     -> G 0.116 -4.953 -0.005
     Node mass:
             Matrix: 14.5125
Node 21      -> B -0.000 -0.116 -0.000
                     -> A 0.116 -4.572 -0.004
                     -> G 0.116 -4.572 -0.004
     Node mass:
             Matrix: 7.2563
Node 22      -> B -0.000 -0.116 -0.000
                     -> A 0.116 -4.191 -0.004
                     -> G 0.116 -4.191 -0.004
     Node mass:
             Matrix: 14.5125
Node 23      -> B -0.000 -0.116 -0.000
                     -> A 0.116 -3.810 -0.003
                     -> G 0.116 -3.810 -0.003
     Node mass:
             Matrix: 7.2563
Node 24      -> B -0.000 -0.116 -0.000
                     -> A 0.116 -3.429 -0.003
                     -> G 0.116 -3.429 -0.003
     Node mass:
             Matrix: 14.5125
Node 25      -> B -0.000 -0.116 -0.000
                     -> A 0.116 -3.048 -0.002
                     -> G 0.116 -3.048 -0.002
     Node mass:
             Matrix: 7.2563
Node 26      -> B -0.000 -0.116 -0.000
                     -> A 0.116 -2.667 -0.002
                     -> G 0.116 -2.667 -0.002
     Node mass:
             Matrix: 14.5125
Node 27      -> B -0.000 -0.116 -0.000
                     -> A 0.116 -2.286 -0.002
                     -> G 0.116 -2.286 -0.002
     Node mass:
             Matrix: 7.2563
Node 28      -> B -0.000 -0.116 -0.000
                     -> A 0.116 -1.905 -0.001
                     -> G 0.116 -1.905 -0.001
     Node mass:
             Matrix: 14.5125
Node 29      -> B -0.000 -0.116 -0.000
                     -> A 0.116 -1.524 -0.001
                     -> G 0.116 -1.524 -0.001
     Node mass:
             Matrix: 7.2563
Node 30      -> B -0.000 -0.116 -0.000
                     -> A 0.116 -1.143 -0.000
                     -> G 0.116 -1.143 -0.000
     Node mass:
             Matrix: 14.5125
Node 31      -> B -0.000 -0.116 -0.000
                     -> A 0.116 -0.762 -0.000
                     -> G 0.116 -0.762 -0.000
     Node mass:
             Matrix: 7.2563
Node 32      -> B -0.000 -0.116 -0.000
                     -> A 0.116 -0.381 -0.000
                     -> G 0.116 -0.381 -0.000
     Node mass:
             Matrix: 14.5125
        Updated the beam C, modal C and K matrices with the terms from the
gravity linearisation

Scaling beam according to reduced time...
     Setting the beam time step to (0.1250)
Updating C and K matrices and natural frequencies with new normalised time...
Model Order Reduction in progress...
Moment Matching Krylov Model Reduction
        Construction Algorithm:
             mimo_rational_arnoldi
        Interpolation points:
             sigma = 0.000000 + 0.000000j [rad/s]

        Krylov order:
             r = 6
     Constructing controllability space
     Constructing observability space
     Deflating column 23
     Deflating column 25
     Deflating column 28
     Deflating column 33
ROM is stable
        DT Eigenvalues:
                mu = 0.992076 + -0.000000j
                mu = 0.971409 + -0.000000j
                mu = 0.957534 + -0.038927j
                mu = 0.957534 + 0.038927j
                mu = 0.954926 + -0.072820j
                mu = 0.954926 + 0.072820j
                mu = 0.954282 + -0.000000j
                mu = 0.935272 + 0.000000j
                mu = 0.927865 + 0.092012j
                mu = 0.927865 + -0.092012j
                mu = 0.927075 + -0.066636j
                mu = 0.927075 + 0.066636j
                mu = 0.925588 + 0.038813j
                mu = 0.925588 + -0.038813j
                mu = 0.922836 + -0.004482j
                mu = 0.922836 + 0.004482j
                mu = 0.915491 + -0.025196j
                mu = 0.915491 + 0.025196j
                mu = 0.908662 + 0.053066j
                mu = 0.908662 + -0.053066j
                mu = 0.881127 + -0.056743j
                mu = 0.881127 + 0.056743j
                mu = 0.882642 + -0.000000j
                mu = 0.867726 + -0.000000j
                mu = 0.717587 + 0.263893j
                mu = 0.717587 + -0.263893j
                mu = 0.697184 + 0.000000j
                mu = 0.360807 + 0.000000j
                mu = -0.000018 + -0.002663j
                mu = -0.000018 + 0.002663j
                mu = 0.000155 + -0.000000j
                mu = -0.000154 + 0.000000j

System reduced from order 6656 to
     n = 32 states
...Completed Model Order Reduction in 5.16 s
Aeroelastic system assembled:
     Aerodynamic states: 32
     Structural states: 4
     Total states: 36
     Inputs: 8
     Outputs: 6
Final system is:
State-space system
States: 36
Inputs: 8
Outputs: 6

Generating an instance of AsymptoticStability
Variable reference_velocity has no assigned value in the settings file.
    will default to the value: 1.0
Variable frequency_cutoff has no assigned value in the settings file.
    will default to the value: 0.0
Variable export_eigenvalues has no assigned value in the settings file.
    will default to the value: False
Variable display_root_locus has no assigned value in the settings file.
    will default to the value: False
Variable iterative_eigvals has no assigned value in the settings file.
    will default to the value: False
Variable num_evals has no assigned value in the settings file.
    will default to the value: 200
Variable postprocessors has no assigned value in the settings file.
    will default to the value: []
Variable postprocessors_settings has no assigned value in the settings file.
    will default to the value: {}
Dynamical System Eigenvalues
Calculating eigenvalues using direct method



|==============|==============|==============|==============|==============|==============|==============|
|     mode     |  eval_real   |  eval_imag   | freq_n (Hz)  | freq_d (Hz)  |   damping    |  period (s)  |
|==============|==============|==============|==============|==============|==============|==============|
|      0       |  -0.021637   |  24.315428   |   3.869922   |   3.869921   |   0.000890   |   0.258403   |
|      1       |  -0.021637   |  -24.315428  |   3.869922   |   3.869921   |   0.000890   |   0.258403   |
|      2       |  -0.069601   |  -0.000000   |   0.011077   |   0.000000   |   1.000000   |     inf      |
|      3       |  -0.105223   |  -21.320535  |   3.393310   |   3.393269   |   0.004935   |   0.294701   |
|      4       |  -0.105223   |  21.320535   |   3.393310   |   3.393269   |   0.004935   |   0.294701   |
|      5       |  -0.253787   |  -0.000000   |   0.040391   |   0.000000   |   1.000000   |     inf      |
|      6       |  -0.372425   |   0.355473   |   0.081940   |   0.056575   |   0.723379   |  17.675568   |
|      7       |  -0.372425   |  -0.355473   |   0.081940   |   0.056575   |   0.723379   |  17.675568   |
|      8       |  -0.378143   |   0.665882   |   0.121875   |   0.105978   |   0.493812   |   9.435879   |
|      9       |  -0.378143   |  -0.665882   |   0.121875   |   0.105978   |   0.493812   |   9.435879   |
|      10      |  -0.409413   |  -0.000000   |   0.065160   |   0.000000   |   1.000000   |     inf      |
|      11      |  -0.585461   |   0.000000   |   0.093179   |   0.000000   |   1.000000   |     inf      |
|      12      |  -0.612214   |  -0.864763   |   0.168631   |   0.137631   |   0.577812   |   7.265789   |
|      13      |  -0.612214   |   0.864763   |   0.168631   |   0.137631   |   0.577812   |   7.265789   |
|      14      |  -0.639935   |   0.627769   |   0.142673   |   0.099913   |   0.713860   |  10.008752   |
|      15      |  -0.639935   |  -0.627769   |   0.142673   |   0.099913   |   0.713860   |  10.008752   |
|      16      |  -0.668833   |   0.366654   |   0.121394   |   0.058355   |   0.876881   |  17.136529   |
|      17      |  -0.668833   |  -0.366654   |   0.121394   |   0.058355   |   0.876881   |  17.136529   |
|      18      |  -0.702465   |   0.042487   |   0.112005   |   0.006762   |   0.998176   |  147.883816  |
|      19      |  -0.702465   |  -0.042487   |   0.112005   |   0.006762   |   0.998176   |  147.883816  |
|      20      |  -0.769174   |   0.240726   |   0.128273   |   0.038313   |   0.954353   |  26.100974   |
|      21      |  -0.769174   |  -0.240726   |   0.128273   |   0.038313   |   0.954353   |  26.100974   |
|      22      |  -0.823096   |   0.510355   |   0.154138   |   0.081226   |   0.849886   |  12.311395   |
|      23      |  -0.823096   |  -0.510355   |   0.154138   |   0.081226   |   0.849886   |  12.311395   |
|      24      |  -1.089098   |   0.562639   |   0.195099   |   0.089547   |   0.888447   |  11.167353   |
|      25      |  -1.089098   |  -0.562639   |   0.195099   |   0.089547   |   0.888447   |  11.167353   |
|      26      |  -1.092180   |  -0.000000   |   0.173826   |   0.000000   |   1.000000   |     inf      |
|      27      |  -1.241288   |   0.000000   |   0.197557   |   0.000000   |   1.000000   |     inf      |
|      28      |  -2.348569   |   3.083089   |   0.616840   |   0.490689   |   0.605970   |   2.037951   |
|      29      |  -2.348569   |  -3.083089   |   0.616840   |   0.490689   |   0.605970   |   2.037951   |
|      30      |  -3.155780   |  -0.000000   |   0.502258   |   0.000000   |   1.000000   |     inf      |
|      31      |  -8.918115   |  -0.000000   |   1.419362   |   0.000000   |   1.000000   |     inf      |
|      32      |  -26.700406  |  -27.485500  |   6.098697   |   4.374453   |   0.696788   |   0.228600   |
|      33      |  -28.824123  |   0.000000   |   4.587502   |   0.000000   |   1.000000   |     inf      |
|      34      |  -35.766803  |  -27.485500  |   7.179135   |   4.374453   |   0.792918   |   0.228600   |
|      35      |  -36.676675  |  -0.000000   |   5.837274   |   0.000000   |   1.000000   |     inf      |
Velocity Asymptotic Stability Analysis
Initial velocity: 100.00 m/s
Final velocity: 180.00 m/s
Number of evaluations: 81
Updating C and K matrices and natural frequencies with new normalised time...
LTI     u: 100.00 m/2   max. CT eig. real: -3.925687
        N unstab.: 000
        Unstable aeroelastic natural frequency CT(rad/s):
Updating C and K matrices and natural frequencies with new normalised time...
LTI     u: 101.00 m/2   max. CT eig. real: -3.951244
        N unstab.: 000
        Unstable aeroelastic natural frequency CT(rad/s):
Updating C and K matrices and natural frequencies with new normalised time...
LTI     u: 102.00 m/2   max. CT eig. real: -3.975951
        N unstab.: 000
        Unstable aeroelastic natural frequency CT(rad/s):
Updating C and K matrices and natural frequencies with new normalised time...
LTI     u: 103.00 m/2   max. CT eig. real: -3.999782
        N unstab.: 000
        Unstable aeroelastic natural frequency CT(rad/s):
Updating C and K matrices and natural frequencies with new normalised time...
LTI     u: 104.00 m/2   max. CT eig. real: -4.022709
        N unstab.: 000
        Unstable aeroelastic natural frequency CT(rad/s):
Updating C and K matrices and natural frequencies with new normalised time...
LTI     u: 105.00 m/2   max. CT eig. real: -4.044705
        N unstab.: 000
        Unstable aeroelastic natural frequency CT(rad/s):
Updating C and K matrices and natural frequencies with new normalised time...
LTI     u: 106.00 m/2   max. CT eig. real: -4.065744
        N unstab.: 000
        Unstable aeroelastic natural frequency CT(rad/s):
Updating C and K matrices and natural frequencies with new normalised time...
LTI     u: 107.00 m/2   max. CT eig. real: -4.085798
        N unstab.: 000
        Unstable aeroelastic natural frequency CT(rad/s):
Updating C and K matrices and natural frequencies with new normalised time...
LTI     u: 108.00 m/2   max. CT eig. real: -4.104841
        N unstab.: 000
        Unstable aeroelastic natural frequency CT(rad/s):
Updating C and K matrices and natural frequencies with new normalised time...
LTI     u: 109.00 m/2   max. CT eig. real: -4.122845
        N unstab.: 000
        Unstable aeroelastic natural frequency CT(rad/s):
Updating C and K matrices and natural frequencies with new normalised time...
LTI     u: 110.00 m/2   max. CT eig. real: -4.139781
        N unstab.: 000
        Unstable aeroelastic natural frequency CT(rad/s):
Updating C and K matrices and natural frequencies with new normalised time...
LTI     u: 111.00 m/2   max. CT eig. real: -4.155617
        N unstab.: 000
        Unstable aeroelastic natural frequency CT(rad/s):
Updating C and K matrices and natural frequencies with new normalised time...
LTI     u: 112.00 m/2   max. CT eig. real: -4.170322
        N unstab.: 000
        Unstable aeroelastic natural frequency CT(rad/s):
Updating C and K matrices and natural frequencies with new normalised time...
LTI     u: 113.00 m/2   max. CT eig. real: -4.183860
        N unstab.: 000
        Unstable aeroelastic natural frequency CT(rad/s):
Updating C and K matrices and natural frequencies with new normalised time...
LTI     u: 114.00 m/2   max. CT eig. real: -4.196192
        N unstab.: 000
        Unstable aeroelastic natural frequency CT(rad/s):
Updating C and K matrices and natural frequencies with new normalised time...
LTI     u: 115.00 m/2   max. CT eig. real: -4.207276
        N unstab.: 000
        Unstable aeroelastic natural frequency CT(rad/s):
Updating C and K matrices and natural frequencies with new normalised time...
LTI     u: 116.00 m/2   max. CT eig. real: -4.217065
        N unstab.: 000
        Unstable aeroelastic natural frequency CT(rad/s):
Updating C and K matrices and natural frequencies with new normalised time...
LTI     u: 117.00 m/2   max. CT eig. real: -4.225507
        N unstab.: 000
        Unstable aeroelastic natural frequency CT(rad/s):
Updating C and K matrices and natural frequencies with new normalised time...
LTI     u: 118.00 m/2   max. CT eig. real: -4.232544
        N unstab.: 000
        Unstable aeroelastic natural frequency CT(rad/s):
Updating C and K matrices and natural frequencies with new normalised time...
LTI     u: 119.00 m/2   max. CT eig. real: -4.238111
        N unstab.: 000
        Unstable aeroelastic natural frequency CT(rad/s):
Updating C and K matrices and natural frequencies with new normalised time...
LTI     u: 120.00 m/2   max. CT eig. real: -4.242138
        N unstab.: 000
        Unstable aeroelastic natural frequency CT(rad/s):
Updating C and K matrices and natural frequencies with new normalised time...
LTI     u: 121.00 m/2   max. CT eig. real: -4.244545
        N unstab.: 000
        Unstable aeroelastic natural frequency CT(rad/s):
Updating C and K matrices and natural frequencies with new normalised time...
LTI     u: 122.00 m/2   max. CT eig. real: -4.245246
        N unstab.: 000
        Unstable aeroelastic natural frequency CT(rad/s):
Updating C and K matrices and natural frequencies with new normalised time...
LTI     u: 123.00 m/2   max. CT eig. real: -4.244143
        N unstab.: 000
        Unstable aeroelastic natural frequency CT(rad/s):
Updating C and K matrices and natural frequencies with new normalised time...
LTI     u: 124.00 m/2   max. CT eig. real: -4.241130
        N unstab.: 000
        Unstable aeroelastic natural frequency CT(rad/s):
Updating C and K matrices and natural frequencies with new normalised time...
LTI     u: 125.00 m/2   max. CT eig. real: -4.236092
        N unstab.: 000
        Unstable aeroelastic natural frequency CT(rad/s):
Updating C and K matrices and natural frequencies with new normalised time...
LTI     u: 126.00 m/2   max. CT eig. real: -4.228899
        N unstab.: 000
        Unstable aeroelastic natural frequency CT(rad/s):
Updating C and K matrices and natural frequencies with new normalised time...
LTI     u: 127.00 m/2   max. CT eig. real: -4.219413
        N unstab.: 000
        Unstable aeroelastic natural frequency CT(rad/s):
Updating C and K matrices and natural frequencies with new normalised time...
LTI     u: 128.00 m/2   max. CT eig. real: -4.207482
        N unstab.: 000
        Unstable aeroelastic natural frequency CT(rad/s):
Updating C and K matrices and natural frequencies with new normalised time...
LTI     u: 129.00 m/2   max. CT eig. real: -4.192940
        N unstab.: 000
        Unstable aeroelastic natural frequency CT(rad/s):
Updating C and K matrices and natural frequencies with new normalised time...
LTI     u: 130.00 m/2   max. CT eig. real: -4.175607
        N unstab.: 000
        Unstable aeroelastic natural frequency CT(rad/s):
Updating C and K matrices and natural frequencies with new normalised time...
LTI     u: 131.00 m/2   max. CT eig. real: -4.155291
        N unstab.: 000
        Unstable aeroelastic natural frequency CT(rad/s):
Updating C and K matrices and natural frequencies with new normalised time...
LTI     u: 132.00 m/2   max. CT eig. real: -4.131780
        N unstab.: 000
        Unstable aeroelastic natural frequency CT(rad/s):
Updating C and K matrices and natural frequencies with new normalised time...
LTI     u: 133.00 m/2   max. CT eig. real: -4.104848
        N unstab.: 000
        Unstable aeroelastic natural frequency CT(rad/s):
Updating C and K matrices and natural frequencies with new normalised time...
LTI     u: 134.00 m/2   max. CT eig. real: -4.074252
        N unstab.: 000
        Unstable aeroelastic natural frequency CT(rad/s):
Updating C and K matrices and natural frequencies with new normalised time...
LTI     u: 135.00 m/2   max. CT eig. real: -4.039730
        N unstab.: 000
        Unstable aeroelastic natural frequency CT(rad/s):
Updating C and K matrices and natural frequencies with new normalised time...
LTI     u: 136.00 m/2   max. CT eig. real: -4.001000
        N unstab.: 000
        Unstable aeroelastic natural frequency CT(rad/s):
Updating C and K matrices and natural frequencies with new normalised time...
LTI     u: 137.00 m/2   max. CT eig. real: -3.957763
        N unstab.: 000
        Unstable aeroelastic natural frequency CT(rad/s):
Updating C and K matrices and natural frequencies with new normalised time...
LTI     u: 138.00 m/2   max. CT eig. real: -3.909697
        N unstab.: 000
        Unstable aeroelastic natural frequency CT(rad/s):
Updating C and K matrices and natural frequencies with new normalised time...
LTI     u: 139.00 m/2   max. CT eig. real: -3.856462
        N unstab.: 000
        Unstable aeroelastic natural frequency CT(rad/s):
Updating C and K matrices and natural frequencies with new normalised time...
LTI     u: 140.00 m/2   max. CT eig. real: -3.797697
        N unstab.: 000
        Unstable aeroelastic natural frequency CT(rad/s):
Updating C and K matrices and natural frequencies with new normalised time...
LTI     u: 141.00 m/2   max. CT eig. real: -3.733020
        N unstab.: 000
        Unstable aeroelastic natural frequency CT(rad/s):
Updating C and K matrices and natural frequencies with new normalised time...
LTI     u: 142.00 m/2   max. CT eig. real: -3.662031
        N unstab.: 000
        Unstable aeroelastic natural frequency CT(rad/s):
Updating C and K matrices and natural frequencies with new normalised time...
LTI     u: 143.00 m/2   max. CT eig. real: -3.584314
        N unstab.: 000
        Unstable aeroelastic natural frequency CT(rad/s):
Updating C and K matrices and natural frequencies with new normalised time...
LTI     u: 144.00 m/2   max. CT eig. real: -3.499436
        N unstab.: 000
        Unstable aeroelastic natural frequency CT(rad/s):
Updating C and K matrices and natural frequencies with new normalised time...
LTI     u: 145.00 m/2   max. CT eig. real: -3.406959
        N unstab.: 000
        Unstable aeroelastic natural frequency CT(rad/s):
Updating C and K matrices and natural frequencies with new normalised time...
LTI     u: 146.00 m/2   max. CT eig. real: -3.306437
        N unstab.: 000
        Unstable aeroelastic natural frequency CT(rad/s):
Updating C and K matrices and natural frequencies with new normalised time...
LTI     u: 147.00 m/2   max. CT eig. real: -3.197433
        N unstab.: 000
        Unstable aeroelastic natural frequency CT(rad/s):
Updating C and K matrices and natural frequencies with new normalised time...
LTI     u: 148.00 m/2   max. CT eig. real: -3.079522
        N unstab.: 000
        Unstable aeroelastic natural frequency CT(rad/s):
Updating C and K matrices and natural frequencies with new normalised time...
LTI     u: 149.00 m/2   max. CT eig. real: -2.952307
        N unstab.: 000
        Unstable aeroelastic natural frequency CT(rad/s):
Updating C and K matrices and natural frequencies with new normalised time...
LTI     u: 150.00 m/2   max. CT eig. real: -2.815436
        N unstab.: 000
        Unstable aeroelastic natural frequency CT(rad/s):
Updating C and K matrices and natural frequencies with new normalised time...
LTI     u: 151.00 m/2   max. CT eig. real: -2.668615
        N unstab.: 000
        Unstable aeroelastic natural frequency CT(rad/s):
Updating C and K matrices and natural frequencies with new normalised time...
LTI     u: 152.00 m/2   max. CT eig. real: -2.511634
        N unstab.: 000
        Unstable aeroelastic natural frequency CT(rad/s):
Updating C and K matrices and natural frequencies with new normalised time...
LTI     u: 153.00 m/2   max. CT eig. real: -2.344382
        N unstab.: 000
        Unstable aeroelastic natural frequency CT(rad/s):
Updating C and K matrices and natural frequencies with new normalised time...
LTI     u: 154.00 m/2   max. CT eig. real: -2.166866
        N unstab.: 000
        Unstable aeroelastic natural frequency CT(rad/s):
Updating C and K matrices and natural frequencies with new normalised time...
LTI     u: 155.00 m/2   max. CT eig. real: -1.979231
        N unstab.: 000
        Unstable aeroelastic natural frequency CT(rad/s):
Updating C and K matrices and natural frequencies with new normalised time...
LTI     u: 156.00 m/2   max. CT eig. real: -1.781770
        N unstab.: 000
        Unstable aeroelastic natural frequency CT(rad/s):
Updating C and K matrices and natural frequencies with new normalised time...
LTI     u: 157.00 m/2   max. CT eig. real: -1.574929
        N unstab.: 000
        Unstable aeroelastic natural frequency CT(rad/s):
Updating C and K matrices and natural frequencies with new normalised time...
LTI     u: 158.00 m/2   max. CT eig. real: -1.359301
        N unstab.: 000
        Unstable aeroelastic natural frequency CT(rad/s):
Updating C and K matrices and natural frequencies with new normalised time...
LTI     u: 159.00 m/2   max. CT eig. real: -1.135613
        N unstab.: 000
        Unstable aeroelastic natural frequency CT(rad/s):
Updating C and K matrices and natural frequencies with new normalised time...
LTI     u: 160.00 m/2   max. CT eig. real: -0.904707
        N unstab.: 000
        Unstable aeroelastic natural frequency CT(rad/s):
Updating C and K matrices and natural frequencies with new normalised time...
LTI     u: 161.00 m/2   max. CT eig. real: -0.667505
        N unstab.: 000
        Unstable aeroelastic natural frequency CT(rad/s):
Updating C and K matrices and natural frequencies with new normalised time...
LTI     u: 162.00 m/2   max. CT eig. real: -0.424982
        N unstab.: 000
        Unstable aeroelastic natural frequency CT(rad/s):
Updating C and K matrices and natural frequencies with new normalised time...
LTI     u: 163.00 m/2   max. CT eig. real: -0.178127
        N unstab.: 000
        Unstable aeroelastic natural frequency CT(rad/s):
Updating C and K matrices and natural frequencies with new normalised time...
LTI     u: 164.00 m/2   max. CT eig. real: 0.072086
        N unstab.: 002
        Unstable aeroelastic natural frequency CT(rad/s):       70.64   70.64
Updating C and K matrices and natural frequencies with new normalised time...
LTI     u: 165.00 m/2   max. CT eig. real: 0.324729
        N unstab.: 002
        Unstable aeroelastic natural frequency CT(rad/s):       70.41   70.41
Updating C and K matrices and natural frequencies with new normalised time...
LTI     u: 166.00 m/2   max. CT eig. real: 0.578936
        N unstab.: 002
        Unstable aeroelastic natural frequency CT(rad/s):       70.20   70.20
Updating C and K matrices and natural frequencies with new normalised time...
LTI     u: 167.00 m/2   max. CT eig. real: 0.833925
        N unstab.: 002
        Unstable aeroelastic natural frequency CT(rad/s):       69.99   69.99
Updating C and K matrices and natural frequencies with new normalised time...
LTI     u: 168.00 m/2   max. CT eig. real: 1.088998
        N unstab.: 002
        Unstable aeroelastic natural frequency CT(rad/s):       69.79   69.79
Updating C and K matrices and natural frequencies with new normalised time...
LTI     u: 169.00 m/2   max. CT eig. real: 1.343552
        N unstab.: 002
        Unstable aeroelastic natural frequency CT(rad/s):       69.61   69.61
Updating C and K matrices and natural frequencies with new normalised time...
LTI     u: 170.00 m/2   max. CT eig. real: 1.597068
        N unstab.: 002
        Unstable aeroelastic natural frequency CT(rad/s):       69.43   69.43
Updating C and K matrices and natural frequencies with new normalised time...
LTI     u: 171.00 m/2   max. CT eig. real: 1.849117
        N unstab.: 002
        Unstable aeroelastic natural frequency CT(rad/s):       69.26   69.26
Updating C and K matrices and natural frequencies with new normalised time...
LTI     u: 172.00 m/2   max. CT eig. real: 2.099343
        N unstab.: 002
        Unstable aeroelastic natural frequency CT(rad/s):       69.10   69.10
Updating C and K matrices and natural frequencies with new normalised time...
LTI     u: 173.00 m/2   max. CT eig. real: 2.347461
        N unstab.: 002
        Unstable aeroelastic natural frequency CT(rad/s):       68.94   68.94
Updating C and K matrices and natural frequencies with new normalised time...
LTI     u: 174.00 m/2   max. CT eig. real: 2.593246
        N unstab.: 002
        Unstable aeroelastic natural frequency CT(rad/s):       68.79   68.79
Updating C and K matrices and natural frequencies with new normalised time...
LTI     u: 175.00 m/2   max. CT eig. real: 2.836529
        N unstab.: 002
        Unstable aeroelastic natural frequency CT(rad/s):       68.65   68.65
Updating C and K matrices and natural frequencies with new normalised time...
LTI     u: 176.00 m/2   max. CT eig. real: 3.077180
        N unstab.: 002
        Unstable aeroelastic natural frequency CT(rad/s):       68.51   68.51
Updating C and K matrices and natural frequencies with new normalised time...
LTI     u: 177.00 m/2   max. CT eig. real: 3.315113
        N unstab.: 002
        Unstable aeroelastic natural frequency CT(rad/s):       68.38   68.38
Updating C and K matrices and natural frequencies with new normalised time...
LTI     u: 178.00 m/2   max. CT eig. real: 3.550268
        N unstab.: 002
        Unstable aeroelastic natural frequency CT(rad/s):       68.25   68.25
Updating C and K matrices and natural frequencies with new normalised time...
LTI     u: 179.00 m/2   max. CT eig. real: 3.782616
        N unstab.: 002
        Unstable aeroelastic natural frequency CT(rad/s):       68.13   68.13
Updating C and K matrices and natural frequencies with new normalised time...
LTI     u: 180.00 m/2   max. CT eig. real: 4.012145
        N unstab.: 002
        Unstable aeroelastic natural frequency CT(rad/s):       68.01   68.01
Saving velocity analysis results...
     Successful
FINISHED - Elapsed time = 59.1569176 seconds
FINISHED - CPU process time = 94.3774846 seconds

[17]:

<sharpy.presharpy.presharpy.PreSharpy at 0x7f85b5231850>

Analysis¶

Nonlinear equilibrium¶

The nonlinear equilibrium condition can be visualised and analysed by opening, with Paraview, the files in the /output/<case_name>/aero and /output/<case_name>/beam folders to see the deflection and aerodynamic forces acting

Stability¶

The stability of the Goland wing is now analysed under changing free stream velocity. The aeroelastic system is projected onto 2 structural modes (1st bending and 1st torsion). The two modes are seen quite separated at 100 m/s. As speed is increased, the damping of the torsion mode decreases until it crosses the imaginary axis onto the right hand plane and flutter begins. This flutter mode is a bending-torsion mode, as seen from the natural frequency plot where the frequencies of each coalesce into this mode.

[18]:

file_name = './output/%s/stability/velocity_analysis_min1000_max1800_nvel0081.dat' % case_name

velocity_analysis = np.loadtxt(file_name)
u_inf = velocity_analysis[:, 0]
eigs_r = velocity_analysis[:, 1]
eigs_i = velocity_analysis[:, 2]

[19]:

fig = plt.figure()
plt.scatter(eigs_r, eigs_i, c=u_inf, cmap='Blues')
cbar = plt.colorbar()
cbar.set_label('Free Stream Velocity, $u_\infty$ [m/s]')

plt.grid()
plt.xlim(-10, 10)
plt.ylim(-150, 150)
plt.xlabel('Real Part, $\lambda$ [rad/s]')
plt.ylabel('Imag Part, $\lambda$ [rad/s]');

_images/content_example_notebooks_linear_goland_flutter_43_0.png

[20]:

fig = plt.figure()
natural_frequency = np.sqrt(eigs_r ** 2 + eigs_i ** 2)
damping_ratio = eigs_r / natural_frequency
cond = (eigs_r>-25) * (eigs_r<10) * (natural_frequency<100) # filter unwanted eigenvalues for this plot (mostly aero modes)

plt.scatter(u_inf[cond], damping_ratio[cond], color='k', marker='s', s=9)

plt.grid()
plt.ylim(-0.25, 0.25)
plt.xlabel('Free Stream Velocity, $u_\infty$ [m/s]')
plt.ylabel('Damping Ratio, $\zeta$ [-]');

_images/content_example_notebooks_linear_goland_flutter_44_0.png

[21]:

fig = plt.figure()
cond = (eigs_r>-25) * (eigs_r<10) # filter unwanted eigenvalues for this plot (mostly aero modes)
plt.scatter(u_inf[cond], natural_frequency[cond], color='k', marker='s', s=9)

plt.grid()
plt.ylim(40, 100)
plt.xlabel('Free Stream Velocity, $u_\infty$ [m/s]')
plt.ylabel('Natural Frequency, $\omega_n$ [rad/s]');

_images/content_example_notebooks_linear_goland_flutter_45_0.png

[1]:

%load_ext autoreload
%autoreload 2
%matplotlib inline
%config InlineBackend.figure_format = 'svg'
import numpy as np
import matplotlib.pyplot as plt
from IPython.display import Image

T-Tail HALE Model tutorial¶

The HALE T-Tail model intends to be a representative example of a typical HALE configuration, with high flexibility and aspect-ratio.

A geometry outline and a summary of the beam properties are given next

[2]:

url = 'https://raw.githubusercontent.com/ImperialCollegeLondon/sharpy/dev_doc/docs/source/content/example_notebooks/images/t-tail_geometry.png'
Image(url=url, width=800)

[2]:

[3]:

url = 'https://raw.githubusercontent.com/ImperialCollegeLondon/sharpy/dev_doc/docs/source/content/example_notebooks/images/t-tail_properties.png'
Image(url=url, width=500)

[3]:

This case is included in tests/coupled/simple_HALE/. The generate_hale.py file in that folder is the one that, if executed, generates all the required SHARPy files. This document is a step by step guide to how to process that case.

The T-Tail HALE model is subject to a 20% 1-cos spanwise constant gust.

First, let’s start with importing SHARPy in our Python environment.

[4]:

import sharpy
import sharpy.sharpy_main as sharpy_main

And now the generate_HALE.py file needs to be executed.

[5]:

route_to_case = '../../../../cases/coupled/simple_HALE/'
%run '../../../../cases/coupled/simple_HALE/generate_hale.py'

There should be 3 new files, apart from the original generate_hale.py:

[6]:

!ls ../../../../cases/coupled/simple_HALE/

generate_hale.py  simple_HALE.aero.h5  simple_HALE.sharpy
output            simple_HALE.fem.h5

SHARPy can be run now. In the terminal, doing cd to the simple_HALE folder, the command would look like:

sharpy simple_HALE.sharpy

From a python console with import sharpy already run, the command is:

[7]:

case_data = sharpy_main.main(['', route_to_case + 'simple_HALE.sharpy'])

--------------------------------------------------------------------------------
            ######  ##     ##    ###    ########  ########  ##    ##
           ##    ## ##     ##   ## ##   ##     ## ##     ##  ##  ##
           ##       ##     ##  ##   ##  ##     ## ##     ##   ####
            ######  ######### ##     ## ########  ########     ##
                 ## ##     ## ######### ##   ##   ##           ##
           ##    ## ##     ## ##     ## ##    ##  ##           ##
            ######  ##     ## ##     ## ##     ## ##           ##
--------------------------------------------------------------------------------
Aeroelastics Lab, Aeronautics Department.
    Copyright (c), Imperial College London.
    All rights reserved.
    License available at https://github.com/imperialcollegelondon/sharpy
Running SHARPy from /home/ng213/2TB/pazy_code/pazy-sharpy/lib/sharpy/docs/source/content/example_notebooks
SHARPy being run is in /home/ng213/2TB/pazy_code/pazy-sharpy/lib/sharpy
The branch being run is dev_setting_error
The version and commit hash are: v1.2.1-348-g402e87e-402e87e
SHARPy output folder set
     /home/ng213/2TB/pazy_code/pazy-sharpy/lib/sharpy/cases/coupled/simple_HALE//output//simple_HALE/
Generating an instance of BeamLoader
Variable for_pos has no assigned value in the settings file.
    will default to the value: [0.0, 0, 0]
Generating an instance of AerogridLoader
Variable control_surface_deflection has no assigned value in the settings file.
    will default to the value: []
Variable control_surface_deflection_generator_settings has no assigned value in the settings file.
    will default to the value: {}
Variable dx1 has no assigned value in the settings file.
    will default to the value: -1.0
Variable ndx1 has no assigned value in the settings file.
    will default to the value: 1
Variable r has no assigned value in the settings file.
    will default to the value: 1.0
Variable dxmax has no assigned value in the settings file.
    will default to the value: -1.0
The aerodynamic grid contains 5 surfaces
  Surface 0, M=4, N=16
     Wake 0, M=20, N=16
  Surface 1, M=4, N=16
     Wake 1, M=20, N=16
  Surface 2, M=4, N=8
     Wake 2, M=20, N=8
  Surface 3, M=4, N=8
     Wake 3, M=20, N=8
  Surface 4, M=4, N=8
     Wake 4, M=20, N=8
  In total: 224 bound panels
  In total: 1120 wake panels
  Total number of panels = 1344
Generating an instance of StaticTrim
Variable print_info has no assigned value in the settings file.
    will default to the value: True
Variable max_iter has no assigned value in the settings file.
    will default to the value: 100
Variable fz_tolerance has no assigned value in the settings file.
    will default to the value: 0.01
Variable fx_tolerance has no assigned value in the settings file.
    will default to the value: 0.01
Variable m_tolerance has no assigned value in the settings file.
    will default to the value: 0.01
Variable tail_cs_index has no assigned value in the settings file.
    will default to the value: 0
Variable thrust_nodes has no assigned value in the settings file.
    will default to the value: [0]
Variable initial_angle_eps has no assigned value in the settings file.
    will default to the value: 0.05
Variable initial_thrust_eps has no assigned value in the settings file.
    will default to the value: 2.0
Variable relaxation_factor has no assigned value in the settings file.
    will default to the value: 0.2
Variable save_info has no assigned value in the settings file.
    will default to the value: False
Generating an instance of StaticCoupled
Variable correct_forces_method has no assigned value in the settings file.
    will default to the value: 
Variable runtime_generators has no assigned value in the settings file.
    will default to the value: {}
Generating an instance of NonLinearStatic
Variable newmark_damp has no assigned value in the settings file.
    will default to the value: 0.0001
Variable gravity_dir has no assigned value in the settings file.
    will default to the value: [0.0, 0.0, 1.0]
Variable relaxation_factor has no assigned value in the settings file.
    will default to the value: 0.3
Variable dt has no assigned value in the settings file.
    will default to the value: 0.01
Variable num_steps has no assigned value in the settings file.
    will default to the value: 500
Variable initial_position has no assigned value in the settings file.
    will default to the value: [0. 0. 0.]
Generating an instance of StaticUvlm
Variable iterative_solver has no assigned value in the settings file.
    will default to the value: False
Variable iterative_tol has no assigned value in the settings file.
    will default to the value: 0.0001
Variable iterative_precond has no assigned value in the settings file.
    will default to the value: False
Variable cfl1 has no assigned value in the settings file.
    will default to the value: True
Variable vortex_radius has no assigned value in the settings file.
    will default to the value: 1e-06
Variable vortex_radius_wake_ind has no assigned value in the settings file.
    will default to the value: 1e-06
Variable rbm_vel_g has no assigned value in the settings file.
    will default to the value: [0.0, 0.0, 0.0, 0.0, 0.0, 0.0]
Variable centre_rot_g has no assigned value in the settings file.
    will default to the value: [0.0, 0.0, 0.0]



|==========|==========|==========|==========|==========|==========|==========|==========|==========|==========|
|   iter   |alpha[deg]|elev[deg] |  thrust  |    Fx    |    Fy    |    Fz    |    Mx    |    My    |    Mz    |
|==========|==========|==========|==========|==========|==========|==========|==========|==========|==========|
|    0     |  4.3100  | -2.0800  |  6.1600  | -0.3600  | -0.0000  |  0.8010  |  0.0000  | -0.6935  |  0.0000  |
|    0     |  7.1748  | -4.9448  |  6.1600  | -27.8570 | -0.0000  | 280.1530 |  0.0000  |-105.5351 |  0.0000  |
|    0     |  4.3100  |  0.7848  |  6.1600  | -2.4506  | -0.0000  | 38.7819  |  0.0000  |-377.3661 | -0.0000  |
|    0     |  4.3100  | -2.0800  |  8.1600  | -2.3600  | -0.0000  |  0.8010  |  0.0000  | -0.6935  |  0.0000  |
|    1     |  4.3018  | -2.0771  |  5.8000  |  0.0660  |  0.0000  | -0.2782  | -0.0000  |  0.2446  |  0.0000  |
|    2     |  4.3039  | -2.0778  |  5.8558  | -0.0068  | -0.0000  |  0.0003  |  0.0000  |  0.0008  |  0.0000  |
Generating an instance of BeamLoads
Variable output_file_name has no assigned value in the settings file.
    will default to the value: beam_loads
Generating an instance of AerogridPlot
Variable include_unsteady_applied_forces has no assigned value in the settings file.
    will default to the value: False
Variable name_prefix has no assigned value in the settings file.
    will default to the value: 
Variable include_velocities has no assigned value in the settings file.
    will default to the value: False
Variable include_incidence_angle has no assigned value in the settings file.
    will default to the value: False
Variable num_cores has no assigned value in the settings file.
    will default to the value: 1
Variable vortex_radius has no assigned value in the settings file.
    will default to the value: 1e-06
...Finished
Generating an instance of BeamPlot
Variable include_FoR has no assigned value in the settings file.
    will default to the value: False
Variable include_applied_moments has no assigned value in the settings file.
    will default to the value: True
Variable name_prefix has no assigned value in the settings file.
    will default to the value: 
Variable output_rbm has no assigned value in the settings file.
    will default to the value: True
...Finished
Generating an instance of DynamicCoupled
Variable print_info has no assigned value in the settings file.
    will default to the value: True
Variable structural_substeps has no assigned value in the settings file.
    will default to the value: 0
Variable dynamic_relaxation has no assigned value in the settings file.
    will default to the value: False
Variable controller_id has no assigned value in the settings file.
    will default to the value: {}
Variable controller_settings has no assigned value in the settings file.
    will default to the value: {}
Variable cleanup_previous_solution has no assigned value in the settings file.
    will default to the value: False
Variable steps_without_unsteady_force has no assigned value in the settings file.
    will default to the value: 0
Variable pseudosteps_ramp_unsteady_force has no assigned value in the settings file.
    will default to the value: 0
Variable correct_forces_method has no assigned value in the settings file.
    will default to the value: 
Variable network_settings has no assigned value in the settings file.
    will default to the value: {}
Variable runtime_generators has no assigned value in the settings file.
    will default to the value: {}
Generating an instance of NonLinearDynamicCoupledStep
Variable num_load_steps has no assigned value in the settings file.
    will default to the value: 1
Variable gravity_dir has no assigned value in the settings file.
    will default to the value: [0.0, 0.0, 1.0]
Variable relaxation_factor has no assigned value in the settings file.
    will default to the value: 0.3
Variable balancing has no assigned value in the settings file.
    will default to the value: False
Variable initial_velocity_direction has no assigned value in the settings file.
    will default to the value: [-1.0, 0.0, 0.0]
Generating an instance of StepUvlm
Variable iterative_solver has no assigned value in the settings file.
    will default to the value: False
Variable iterative_tol has no assigned value in the settings file.
    will default to the value: 0.0001
Variable iterative_precond has no assigned value in the settings file.
    will default to the value: False
Variable cfl1 has no assigned value in the settings file.
    will default to the value: True
Variable vortex_radius has no assigned value in the settings file.
    will default to the value: 1e-06
Variable vortex_radius_wake_ind has no assigned value in the settings file.
    will default to the value: 1e-06
Variable interp_coords has no assigned value in the settings file.
    will default to the value: 0
Variable filter_method has no assigned value in the settings file.
    will default to the value: 0
Variable interp_method has no assigned value in the settings file.
    will default to the value: 0
Variable yaw_slerp has no assigned value in the settings file.
    will default to the value: 0.0
Variable centre_rot has no assigned value in the settings file.
    will default to the value: [0.0, 0.0, 0.0]
Variable quasi_steady has no assigned value in the settings file.
    will default to the value: False
gamma_dot_filtering does not support even numbers.Changing 6 to 7
Generating an instance of BeamLoads
Variable output_file_name has no assigned value in the settings file.
    will default to the value: beam_loads
Generating an instance of BeamPlot
Variable include_FoR has no assigned value in the settings file.
    will default to the value: False
Variable include_applied_moments has no assigned value in the settings file.
    will default to the value: True
Variable name_prefix has no assigned value in the settings file.
    will default to the value: 
Variable output_rbm has no assigned value in the settings file.
    will default to the value: True
Generating an instance of AerogridPlot
Variable include_forward_motion has no assigned value in the settings file.
    will default to the value: False
Variable include_unsteady_applied_forces has no assigned value in the settings file.
    will default to the value: False
Variable name_prefix has no assigned value in the settings file.
    will default to the value: 
Variable u_inf has no assigned value in the settings file.
    will default to the value: 0.0
Variable dt has no assigned value in the settings file.
    will default to the value: 0.0
Variable include_velocities has no assigned value in the settings file.
    will default to the value: False
Variable include_incidence_angle has no assigned value in the settings file.
    will default to the value: False
Variable num_cores has no assigned value in the settings file.
    will default to the value: 1
Variable vortex_radius has no assigned value in the settings file.
    will default to the value: 1e-06



|=======|========|======|==============|==============|==============|==============|==============|
|  ts   |   t    | iter | struc ratio  |  iter time   | residual vel |  FoR_vel(x)  |  FoR_vel(z)  |
|=======|========|======|==============|==============|==============|==============|==============|

/home/ng213/2TB/pazy_code/pazy-sharpy/lib/sharpy/sharpy/aero/utils/uvlmlib.py:264: RuntimeWarning: invalid value encountered in true_divide
  flightconditions.uinf_direction = np.ctypeslib.as_ctypes(ts_info.u_ext[0][:, 0, 0]/flightconditions.uinf)
/home/ng213/2TB/pazy_code/pazy-sharpy/lib/sharpy/sharpy/aero/utils/uvlmlib.py:325: RuntimeWarning: invalid value encountered in true_divide
  flightconditions.uinf_direction = np.ctypeslib.as_ctypes(ts_info.u_ext[0][:, 0, 0]/flightconditions.uinf)

|   1   | 0.0250 |  2   |   0.660158   |   0.788767   |  -4.245993   |-9.971789e+00 |-7.504783e-01 |
|   2   | 0.0500 |  3   |   0.698975   |   0.965136   |  -4.635143   |-9.971745e+00 |-7.505379e-01 |
|   3   | 0.0750 |  2   |   0.699509   |   0.744684   |  -4.142638   |-9.971678e+00 |-7.506741e-01 |
|   4   | 0.1000 |  2   |   0.702553   |   0.742001   |  -4.583042   |-9.971654e+00 |-7.508442e-01 |
|   5   | 0.1250 |  1   |   0.705584   |   0.512845   |  -4.384776   |-9.971729e+00 |-7.509873e-01 |
|   6   | 0.1500 |  2   |   0.710967   |   0.733027   |  -4.729199   |-9.971808e+00 |-7.510934e-01 |
|   7   | 0.1750 |  1   |   0.717295   |   0.511901   |  -4.049012   |-9.971749e+00 |-7.511575e-01 |
|   8   | 0.2000 |  1   |   0.687752   |   0.519393   |  -4.147149   |-9.971609e+00 |-7.511620e-01 |
|   9   | 0.2250 |  1   |   0.657027   |   0.533608   |  -4.243231   |-9.971483e+00 |-7.511461e-01 |
|  10   | 0.2500 |  1   |   0.650443   |   0.540742   |  -4.352169   |-9.971378e+00 |-7.511644e-01 |
|  11   | 0.2750 |  1   |   0.656218   |   0.534101   |  -4.191190   |-9.971371e+00 |-7.511728e-01 |
|  12   | 0.3000 |  1   |   0.671768   |   0.554114   |  -4.122823   |-9.971442e+00 |-7.511539e-01 |
|  13   | 0.3250 |  1   |   0.656158   |   0.537498   |  -4.418002   |-9.971429e+00 |-7.511300e-01 |
|  14   | 0.3500 |  1   |   0.704366   |   0.508580   |  -4.678378   |-9.971349e+00 |-7.510900e-01 |
|  15   | 0.3750 |  1   |   0.719952   |   0.514705   |  -4.402185   |-9.971299e+00 |-7.510668e-01 |
|  16   | 0.4000 |  1   |   0.705019   |   0.512428   |  -4.448460   |-9.971290e+00 |-7.510571e-01 |
|  17   | 0.4250 |  1   |   0.713898   |   0.511336   |  -4.476767   |-9.971358e+00 |-7.510533e-01 |
|  18   | 0.4500 |  1   |   0.715516   |   0.511316   |  -4.534268   |-9.971488e+00 |-7.510753e-01 |
|  19   | 0.4750 |  1   |   0.709680   |   0.508300   |  -4.662816   |-9.971575e+00 |-7.510784e-01 |
|  20   | 0.5000 |  2   |   0.704182   |   0.744397   |  -4.812376   |-9.971586e+00 |-7.510241e-01 |
|  21   | 0.5250 |  3   |   0.708522   |   0.965943   |  -4.402757   |-9.971726e+00 |-7.507397e-01 |
|  22   | 0.5500 |  4   |   0.708365   |   1.200794   |  -4.475885   |-9.972374e+00 |-7.497152e-01 |
|  23   | 0.5750 |  4   |   0.708962   |   1.199187   |  -4.264700   |-9.973956e+00 |-7.471042e-01 |
|  24   | 0.6000 |  4   |   0.709056   |   1.201371   |  -4.148547   |-9.977005e+00 |-7.417512e-01 |
|  25   | 0.6250 |  4   |   0.707360   |   1.214448   |  -4.101431   |-9.982064e+00 |-7.323149e-01 |
|  26   | 0.6500 |  4   |   0.707825   |   1.211973   |  -4.083226   |-9.989506e+00 |-7.173484e-01 |
|  27   | 0.6750 |  4   |   0.705636   |   1.209574   |  -4.081045   |-9.999533e+00 |-6.954412e-01 |
|  28   | 0.7000 |  4   |   0.702970   |   1.199970   |  -4.095091   |-1.001216e+01 |-6.653667e-01 |
|  29   | 0.7250 |  4   |   0.705531   |   1.206136   |  -4.120431   |-1.002734e+01 |-6.261403e-01 |
|  30   | 0.7500 |  4   |   0.702935   |   1.213436   |  -4.153507   |-1.004515e+01 |-5.771350e-01 |
|  31   | 0.7750 |  4   |   0.698230   |   1.203890   |  -4.194909   |-1.006573e+01 |-5.180671e-01 |
|  32   | 0.8000 |  4   |   0.703664   |   1.208508   |  -4.236529   |-1.008907e+01 |-4.489843e-01 |
|  33   | 0.8250 |  4   |   0.701559   |   1.203176   |  -4.281618   |-1.011502e+01 |-3.702984e-01 |
|  34   | 0.8500 |  4   |   0.700114   |   1.215140   |  -4.272027   |-1.014323e+01 |-2.827796e-01 |
|  35   | 0.8750 |  4   |   0.701394   |   1.206869   |  -4.137638   |-1.017318e+01 |-1.875441e-01 |
|  36   | 0.9000 |  4   |   0.698019   |   1.213434   |  -4.087979   |-1.020434e+01 |-8.606667e-02 |
|  37   | 0.9250 |  4   |   0.697820   |   1.213205   |  -4.081173   |-1.023614e+01 | 1.986601e-02 |
|  38   | 0.9500 |  4   |   0.692925   |   1.210431   |  -4.067952   |-1.026798e+01 | 1.282642e-01 |
|  39   | 0.9750 |  4   |   0.692719   |   1.203242   |  -4.040392   |-1.029913e+01 | 2.371167e-01 |
|  40   | 1.0000 |  4   |   0.695089   |   1.205393   |  -4.023531   |-1.032875e+01 | 3.445071e-01 |
|  41   | 1.0250 |  5   |   0.696443   |   1.438348   |  -4.548408   |-1.035594e+01 | 4.482817e-01 |
|  42   | 1.0500 |  5   |   0.699286   |   1.456647   |  -4.496733   |-1.037989e+01 | 5.458046e-01 |
|  43   | 1.0750 |  5   |   0.693181   |   1.451422   |  -4.426046   |-1.040006e+01 | 6.342879e-01 |
|  44   | 1.1000 |  5   |   0.672004   |   1.475300   |  -4.379701   |-1.041606e+01 | 7.112695e-01 |
|  45   | 1.1250 |  5   |   0.672251   |   1.468131   |  -4.402469   |-1.042753e+01 | 7.747229e-01 |
|  46   | 1.1500 |  5   |   0.689079   |   1.445732   |  -4.477108   |-1.043429e+01 | 8.230013e-01 |
|  47   | 1.1750 |  5   |   0.690835   |   1.446807   |  -4.548994   |-1.043627e+01 | 8.549797e-01 |
|  48   | 1.2000 |  5   |   0.692859   |   1.458130   |  -4.602614   |-1.043347e+01 | 8.701306e-01 |
|  49   | 1.2250 |  5   |   0.692201   |   1.450715   |  -4.670662   |-1.042601e+01 | 8.683187e-01 |
|  50   | 1.2500 |  4   |   0.692043   |   1.211332   |  -4.009458   |-1.041409e+01 | 8.498052e-01 |
|  51   | 1.2750 |  5   |   0.691406   |   1.450458   |  -4.437774   |-1.039785e+01 | 8.153184e-01 |
|  52   | 1.3000 |  5   |   0.690078   |   1.456433   |  -4.400817   |-1.037756e+01 | 7.658481e-01 |
|  53   | 1.3250 |  5   |   0.642589   |   1.516374   |  -4.491993   |-1.035352e+01 | 7.027470e-01 |
|  54   | 1.3500 |  5   |   0.636213   |   1.529562   |  -4.660919   |-1.032602e+01 | 6.277602e-01 |
|  55   | 1.3750 |  4   |   0.654913   |   1.257229   |  -4.127066   |-1.029540e+01 | 5.429281e-01 |
|  56   | 1.4000 |  4   |   0.690183   |   1.213338   |  -4.180862   |-1.026211e+01 | 4.504884e-01 |
|  57   | 1.4250 |  4   |   0.690784   |   1.220024   |  -4.119756   |-1.022658e+01 | 3.526569e-01 |
|  58   | 1.4500 |  4   |   0.693190   |   1.230164   |  -4.115756   |-1.018920e+01 | 2.516427e-01 |
|  59   | 1.4750 |  4   |   0.689614   |   1.227466   |  -4.089544   |-1.015042e+01 | 1.494680e-01 |
|  60   | 1.5000 |  4   |   0.686766   |   1.221660   |  -4.073221   |-1.011070e+01 | 4.789509e-02 |
|  61   | 1.5250 |  4   |   0.688439   |   1.225181   |  -4.065044   |-1.007052e+01 |-5.142849e-02 |
|  62   | 1.5500 |  4   |   0.686541   |   1.221131   |  -4.066648   |-1.003034e+01 |-1.468432e-01 |
|  63   | 1.5750 |  4   |   0.649074   |   1.269069   |  -4.116776   |-9.990515e+00 |-2.367221e-01 |
|  64   | 1.6000 |  4   |   0.634231   |   1.298771   |  -4.232749   |-9.951260e+00 |-3.197660e-01 |
|  65   | 1.6250 |  4   |   0.631620   |   1.295939   |  -4.346372   |-9.912713e+00 |-3.949816e-01 |
|  66   | 1.6500 |  4   |   0.676708   |   1.241231   |  -4.462283   |-9.875008e+00 |-4.616848e-01 |
|  67   | 1.6750 |  4   |   0.690826   |   1.236500   |  -4.536211   |-9.838289e+00 |-5.194903e-01 |
|  68   | 1.7000 |  4   |   0.685831   |   1.222472   |  -4.557395   |-9.802717e+00 |-5.683337e-01 |
|  69   | 1.7250 |  4   |   0.683209   |   1.225972   |  -4.514445   |-9.768493e+00 |-6.084703e-01 |
|  70   | 1.7500 |  4   |   0.681402   |   1.219586   |  -4.515767   |-9.735644e+00 |-6.403299e-01 |
|  71   | 1.7750 |  4   |   0.679701   |   1.241776   |  -4.561782   |-9.704041e+00 |-6.644313e-01 |
|  72   | 1.8000 |  4   |   0.687585   |   1.233221   |  -4.576978   |-9.673585e+00 |-6.813177e-01 |
|  73   | 1.8250 |  3   |   0.685128   |   0.992525   |  -4.085489   |-9.644153e+00 |-6.916357e-01 |
|  74   | 1.8500 |  3   |   0.630406   |   1.081735   |  -4.228827   |-9.615525e+00 |-6.960345e-01 |
|  75   | 1.8750 |  3   |   0.633953   |   1.068277   |  -4.281040   |-9.587600e+00 |-6.952011e-01 |
|  76   | 1.9000 |  3   |   0.631463   |   1.060051   |  -4.298619   |-9.560401e+00 |-6.899257e-01 |
|  77   | 1.9250 |  3   |   0.641269   |   1.076329   |  -4.263283   |-9.533905e+00 |-6.810428e-01 |
|  78   | 1.9500 |  3   |   0.632168   |   1.056881   |  -4.172553   |-9.508183e+00 |-6.694250e-01 |
|  79   | 1.9750 |  3   |   0.644723   |   1.078860   |  -4.087294   |-9.483448e+00 |-6.559401e-01 |
|  80   | 2.0000 |  3   |   0.629024   |   1.058206   |  -4.017983   |-9.459903e+00 |-6.414764e-01 |
|  81   | 2.0250 |  4   |   0.641541   |   1.337528   |  -4.489993   |-9.437808e+00 |-6.269365e-01 |
|  82   | 2.0500 |  4   |   0.628707   |   1.310679   |  -4.470211   |-9.417538e+00 |-6.131778e-01 |
|  83   | 2.0750 |  4   |   0.628663   |   1.307533   |  -4.453642   |-9.399441e+00 |-6.009539e-01 |
|  84   | 2.1000 |  4   |   0.631778   |   1.301269   |  -4.451153   |-9.383802e+00 |-5.909513e-01 |
|  85   | 2.1250 |  4   |   0.629938   |   1.308679   |  -4.454054   |-9.370919e+00 |-5.837527e-01 |
|  86   | 2.1500 |  4   |   0.630758   |   1.309061   |  -4.489244   |-9.361058e+00 |-5.798117e-01 |
|  87   | 2.1750 |  4   |   0.630606   |   1.308830   |  -4.529495   |-9.354357e+00 |-5.794602e-01 |
|  88   | 2.2000 |  4   |   0.629895   |   1.306216   |  -4.555062   |-9.350910e+00 |-5.828920e-01 |
|  89   | 2.2250 |  4   |   0.627658   |   1.299544   |  -4.583501   |-9.350758e+00 |-5.901807e-01 |
|  90   | 2.2500 |  4   |   0.631380   |   1.304904   |  -4.577435   |-9.353747e+00 |-6.012367e-01 |
|  91   | 2.2750 |  3   |   0.632640   |   1.049719   |  -4.015225   |-9.359659e+00 |-6.158432e-01 |
|  92   | 2.3000 |  3   |   0.677727   |   0.998599   |  -4.029324   |-9.368239e+00 |-6.336696e-01 |
|  93   | 2.3250 |  3   |   0.681171   |   0.994001   |  -4.027658   |-9.379037e+00 |-6.542748e-01 |
|  94   | 2.3500 |  3   |   0.681259   |   0.989943   |  -4.032139   |-9.391519e+00 |-6.771019e-01 |
|  95   | 2.3750 |  3   |   0.662018   |   1.012158   |  -4.018542   |-9.405213e+00 |-7.015133e-01 |
|  96   | 2.4000 |  4   |   0.649296   |   1.280039   |  -4.507534   |-9.419633e+00 |-7.269046e-01 |
|  97   | 2.4250 |  4   |   0.643727   |   1.278447   |  -4.486197   |-9.434238e+00 |-7.525958e-01 |
|  98   | 2.4500 |  4   |   0.626945   |   1.300289   |  -4.469752   |-9.448552e+00 |-7.778583e-01 |
|  99   | 2.4750 |  4   |   0.630502   |   1.316456   |  -4.479896   |-9.462111e+00 |-8.020210e-01 |
|  100  | 2.5000 |  4   |   0.624389   |   1.310294   |  -4.502345   |-9.474427e+00 |-8.243816e-01 |
|  101  | 2.5250 |  4   |   0.631037   |   1.315843   |  -4.533404   |-9.485161e+00 |-8.442818e-01 |
|  102  | 2.5500 |  4   |   0.644611   |   1.283856   |  -4.554463   |-9.494088e+00 |-8.611748e-01 |
|  103  | 2.5750 |  4   |   0.680174   |   1.237591   |  -4.579594   |-9.501053e+00 |-8.746547e-01 |
|  104  | 2.6000 |  4   |   0.680077   |   1.236782   |  -4.596771   |-9.506032e+00 |-8.844994e-01 |
|  105  | 2.6250 |  4   |   0.675584   |   1.232872   |  -4.484950   |-9.509125e+00 |-8.907008e-01 |
|  106  | 2.6500 |  4   |   0.678916   |   1.235079   |  -4.463714   |-9.510468e+00 |-8.934511e-01 |
|  107  | 2.6750 |  4   |   0.677563   |   1.234071   |  -4.381520   |-9.510274e+00 |-8.931021e-01 |
|  108  | 2.7000 |  4   |   0.674451   |   1.246872   |  -4.443680   |-9.508969e+00 |-8.901755e-01 |
|  109  | 2.7250 |  4   |   0.680008   |   1.243277   |  -4.646030   |-9.507028e+00 |-8.852282e-01 |
|  110  | 2.7500 |  3   |   0.687824   |   1.013889   |  -4.095041   |-9.504803e+00 |-8.787479e-01 |
|  111  | 2.7750 |  3   |   0.678555   |   1.000272   |  -4.324971   |-9.502690e+00 |-8.711948e-01 |
|  112  | 2.8000 |  3   |   0.629410   |   1.056518   |  -4.391682   |-9.501123e+00 |-8.630844e-01 |
|  113  | 2.8250 |  3   |   0.631289   |   1.061790   |  -4.405523   |-9.500417e+00 |-8.548953e-01 |
|  114  | 2.8500 |  3   |   0.623451   |   1.061574   |  -4.405133   |-9.500842e+00 |-8.469691e-01 |
|  115  | 2.8750 |  3   |   0.629521   |   1.063894   |  -4.364937   |-9.502651e+00 |-8.395888e-01 |
|  116  | 2.9000 |  3   |   0.625789   |   1.052592   |  -4.333192   |-9.505900e+00 |-8.329495e-01 |
|  117  | 2.9250 |  3   |   0.634558   |   1.059645   |  -4.287009   |-9.510496e+00 |-8.271466e-01 |
|  118  | 2.9500 |  3   |   0.676550   |   1.015943   |  -4.245942   |-9.516385e+00 |-8.222099e-01 |
|  119  | 2.9750 |  3   |   0.680614   |   1.002676   |  -4.205825   |-9.523418e+00 |-8.181140e-01 |
|  120  | 3.0000 |  3   |   0.682271   |   1.006870   |  -4.188187   |-9.531318e+00 |-8.148053e-01 |
|  121  | 3.0250 |  3   |   0.680266   |   0.996102   |  -4.232299   |-9.539814e+00 |-8.121677e-01 |
|  122  | 3.0500 |  3   |   0.681732   |   0.999955   |  -4.231690   |-9.548602e+00 |-8.100300e-01 |
|  123  | 3.0750 |  3   |   0.682858   |   1.010936   |  -4.279188   |-9.557348e+00 |-8.082342e-01 |
|  124  | 3.1000 |  3   |   0.681190   |   1.009030   |  -4.271080   |-9.565764e+00 |-8.066818e-01 |
|  125  | 3.1250 |  3   |   0.683038   |   1.007188   |  -4.392752   |-9.573620e+00 |-8.052156e-01 |
|  126  | 3.1500 |  3   |   0.678701   |   1.000200   |  -4.449904   |-9.580739e+00 |-8.036619e-01 |
|  127  | 3.1750 |  2   |   0.676607   |   0.762312   |  -4.000854   |-9.587007e+00 |-8.019792e-01 |
|  128  | 3.2000 |  3   |   0.676056   |   1.006674   |  -4.538007   |-9.592408e+00 |-8.001705e-01 |
|  129  | 3.2250 |  3   |   0.675263   |   1.005720   |  -4.547103   |-9.596953e+00 |-7.981922e-01 |
|  130  | 3.2500 |  3   |   0.676868   |   1.006415   |  -4.525018   |-9.600665e+00 |-7.960410e-01 |
|  131  | 3.2750 |  3   |   0.675184   |   1.010851   |  -4.506300   |-9.603640e+00 |-7.937587e-01 |
|  132  | 3.3000 |  3   |   0.672328   |   1.002885   |  -4.509537   |-9.606035e+00 |-7.913608e-01 |
|  133  | 3.3250 |  3   |   0.671273   |   1.006276   |  -4.517757   |-9.608054e+00 |-7.889411e-01 |
|  134  | 3.3500 |  2   |   0.667902   |   0.767805   |  -4.004169   |-9.609914e+00 |-7.866119e-01 |
|  135  | 3.3750 |  2   |   0.672012   |   0.762166   |  -4.037206   |-9.611796e+00 |-7.844475e-01 |
|  136  | 3.4000 |  2   |   0.674040   |   0.766492   |  -4.091199   |-9.613836e+00 |-7.825251e-01 |
|  137  | 3.4250 |  2   |   0.677833   |   0.764682   |  -4.161763   |-9.616142e+00 |-7.808889e-01 |
|  138  | 3.4500 |  2   |   0.678520   |   0.772371   |  -4.208416   |-9.618837e+00 |-7.795641e-01 |
|  139  | 3.4750 |  2   |   0.675557   |   0.769200   |  -4.185191   |-9.622012e+00 |-7.786017e-01 |
|  140  | 3.5000 |  2   |   0.658891   |   0.777941   |  -4.143336   |-9.625717e+00 |-7.780201e-01 |
|  141  | 3.5250 |  2   |   0.670239   |   0.764967   |  -4.111505   |-9.629952e+00 |-7.777878e-01 |
|  142  | 3.5500 |  2   |   0.675294   |   0.780672   |  -4.095589   |-9.634644e+00 |-7.778791e-01 |
|  143  | 3.5750 |  2   |   0.674070   |   0.768821   |  -4.089987   |-9.639706e+00 |-7.782389e-01 |
|  144  | 3.6000 |  2   |   0.675235   |   0.770859   |  -4.092880   |-9.645044e+00 |-7.787973e-01 |
|  145  | 3.6250 |  2   |   0.667235   |   0.766770   |  -4.113065   |-9.650579e+00 |-7.794970e-01 |
|  146  | 3.6500 |  2   |   0.664948   |   0.775005   |  -4.162262   |-9.656221e+00 |-7.802848e-01 |
|  147  | 3.6750 |  2   |   0.669905   |   0.778505   |  -4.233073   |-9.661859e+00 |-7.810839e-01 |
|  148  | 3.7000 |  2   |   0.673135   |   0.772653   |  -4.300425   |-9.667393e+00 |-7.818259e-01 |
|  149  | 3.7250 |  2   |   0.668814   |   0.767011   |  -4.341062   |-9.672713e+00 |-7.824584e-01 |
|  150  | 3.7500 |  2   |   0.667366   |   0.768942   |  -4.347827   |-9.677758e+00 |-7.829222e-01 |
|  151  | 3.7750 |  2   |   0.671508   |   0.768175   |  -4.340552   |-9.682512e+00 |-7.832059e-01 |
|  152  | 3.8000 |  2   |   0.657508   |   0.783694   |  -4.331623   |-9.686971e+00 |-7.832953e-01 |
|  153  | 3.8250 |  2   |   0.672415   |   0.768370   |  -4.316722   |-9.691148e+00 |-7.831777e-01 |
|  154  | 3.8500 |  2   |   0.677129   |   0.779227   |  -4.299262   |-9.695058e+00 |-7.828747e-01 |
|  155  | 3.8750 |  2   |   0.673584   |   0.770337   |  -4.305853   |-9.698731e+00 |-7.823999e-01 |
|  156  | 3.9000 |  2   |   0.665485   |   0.772028   |  -4.334291   |-9.702215e+00 |-7.817778e-01 |
|  157  | 3.9250 |  2   |   0.666994   |   0.776969   |  -4.377407   |-9.705587e+00 |-7.810551e-01 |
|  158  | 3.9500 |  2   |   0.665925   |   0.780606   |  -4.439104   |-9.708933e+00 |-7.802775e-01 |
|  159  | 3.9750 |  2   |   0.670904   |   0.771282   |  -4.498475   |-9.712316e+00 |-7.794772e-01 |
|  160  | 4.0000 |  1   |   0.671639   |   0.530210   |  -4.008078   |-9.715787e+00 |-7.786931e-01 |
|  161  | 4.0250 |  1   |   0.670327   |   0.533749   |  -4.092355   |-9.719380e+00 |-7.779506e-01 |
|  162  | 4.0500 |  1   |   0.668751   |   0.537579   |  -4.156978   |-9.723116e+00 |-7.772698e-01 |
|  163  | 4.0750 |  1   |   0.663677   |   0.531485   |  -4.072783   |-9.727029e+00 |-7.766788e-01 |
|  164  | 4.1000 |  2   |   0.666350   |   0.773267   |  -4.569537   |-9.731146e+00 |-7.761920e-01 |
|  165  | 4.1250 |  2   |   0.682839   |   0.788505   |  -4.570174   |-9.735472e+00 |-7.758138e-01 |
|  166  | 4.1500 |  2   |   0.620765   |   0.812113   |  -4.549853   |-9.739981e+00 |-7.755403e-01 |
|  167  | 4.1750 |  2   |   0.618753   |   0.825905   |  -4.537868   |-9.744625e+00 |-7.753472e-01 |
|  168  | 4.2000 |  2   |   0.618510   |   0.819368   |  -4.524976   |-9.749356e+00 |-7.752100e-01 |
|  169  | 4.2250 |  2   |   0.613543   |   0.817032   |  -4.515910   |-9.754134e+00 |-7.751152e-01 |
|  170  | 4.2500 |  2   |   0.614536   |   0.814042   |  -4.538699   |-9.758943e+00 |-7.750408e-01 |
|  171  | 4.2750 |  1   |   0.625472   |   0.574294   |  -4.020787   |-9.763759e+00 |-7.749700e-01 |
|  172  | 4.3000 |  1   |   0.608604   |   0.560485   |  -4.061045   |-9.768537e+00 |-7.748838e-01 |
|  173  | 4.3250 |  1   |   0.613070   |   0.563931   |  -4.121220   |-9.773242e+00 |-7.747597e-01 |
|  174  | 4.3500 |  1   |   0.642887   |   0.551584   |  -4.110854   |-9.777831e+00 |-7.745740e-01 |
|  175  | 4.3750 |  1   |   0.666433   |   0.534772   |  -4.088306   |-9.782291e+00 |-7.743221e-01 |
|  176  | 4.4000 |  1   |   0.673361   |   0.543510   |  -4.077439   |-9.786631e+00 |-7.740007e-01 |
|  177  | 4.4250 |  1   |   0.675407   |   0.545999   |  -4.085008   |-9.790868e+00 |-7.736116e-01 |
|  178  | 4.4500 |  1   |   0.663487   |   0.537923   |  -4.084174   |-9.795016e+00 |-7.731651e-01 |
|  179  | 4.4750 |  1   |   0.656179   |   0.541705   |  -4.095747   |-9.799077e+00 |-7.726600e-01 |
|  180  | 4.5000 |  1   |   0.665239   |   0.538480   |  -4.143039   |-9.803063e+00 |-7.721067e-01 |
|  181  | 4.5250 |  1   |   0.671591   |   0.542443   |  -4.219653   |-9.806995e+00 |-7.715207e-01 |
|  182  | 4.5500 |  1   |   0.663922   |   0.536485   |  -4.324315   |-9.810904e+00 |-7.709163e-01 |
|  183  | 4.5750 |  1   |   0.657884   |   0.540013   |  -4.462694   |-9.814833e+00 |-7.703121e-01 |
|  184  | 4.6000 |  1   |   0.664151   |   0.536879   |  -4.555594   |-9.818801e+00 |-7.697261e-01 |
|  185  | 4.6250 |  1   |   0.664279   |   0.534583   |  -4.636689   |-9.822816e+00 |-7.691634e-01 |
|  186  | 4.6500 |  1   |   0.663570   |   0.535142   |  -4.623202   |-9.826877e+00 |-7.686317e-01 |
|  187  | 4.6750 |  1   |   0.654862   |   0.535068   |  -4.465982   |-9.830990e+00 |-7.681383e-01 |
|  188  | 4.7000 |  1   |   0.664028   |   0.540949   |  -4.338112   |-9.835165e+00 |-7.676893e-01 |
|  189  | 4.7250 |  1   |   0.663004   |   0.538335   |  -4.294846   |-9.839409e+00 |-7.672874e-01 |
|  190  | 4.7500 |  1   |   0.661509   |   0.535572   |  -4.283904   |-9.843723e+00 |-7.669317e-01 |
|  191  | 4.7750 |  1   |   0.667410   |   0.544977   |  -4.289664   |-9.848084e+00 |-7.666140e-01 |
|  192  | 4.8000 |  1   |   0.631005   |   0.557035   |  -4.285048   |-9.852468e+00 |-7.663236e-01 |
|  193  | 4.8250 |  1   |   0.655634   |   0.543174   |  -4.249586   |-9.856861e+00 |-7.660553e-01 |
|  194  | 4.8500 |  1   |   0.663471   |   0.540511   |  -4.240242   |-9.861248e+00 |-7.657973e-01 |
|  195  | 4.8750 |  1   |   0.663061   |   0.537256   |  -4.279527   |-9.865630e+00 |-7.655457e-01 |
|  196  | 4.9000 |  1   |   0.666065   |   0.559327   |  -4.341423   |-9.869997e+00 |-7.652934e-01 |
|  197  | 4.9250 |  1   |   0.655532   |   0.540197   |  -4.389727   |-9.874334e+00 |-7.650274e-01 |
|  198  | 4.9500 |  1   |   0.653665   |   0.536703   |  -4.392163   |-9.878621e+00 |-7.647419e-01 |
|  199  | 4.9750 |  1   |   0.643891   |   0.551159   |  -4.356375   |-9.882848e+00 |-7.644317e-01 |
|  200  | 5.0000 |  1   |   0.658182   |   0.536739   |  -4.357632   |-9.887023e+00 |-7.640960e-01 |
|  201  | 5.0250 |  1   |   0.666568   |   0.547456   |  -4.374468   |-9.891152e+00 |-7.637354e-01 |
|  202  | 5.0500 |  1   |   0.690858   |   0.576716   |  -4.405701   |-9.895246e+00 |-7.633546e-01 |
|  203  | 5.0750 |  1   |   0.648976   |   0.543039   |  -4.457054   |-9.899310e+00 |-7.629532e-01 |
|  204  | 5.1000 |  1   |   0.645745   |   0.541463   |  -4.487556   |-9.903338e+00 |-7.625328e-01 |
|  205  | 5.1250 |  1   |   0.591463   |   0.597350   |  -4.532516   |-9.907334e+00 |-7.620999e-01 |
|  206  | 5.1500 |  1   |   0.602864   |   0.567360   |  -4.584225   |-9.911313e+00 |-7.616581e-01 |
|  207  | 5.1750 |  1   |   0.607332   |   0.564882   |  -4.653390   |-9.915290e+00 |-7.612183e-01 |
|  208  | 5.2000 |  1   |   0.617578   |   0.575872   |  -4.738979   |-9.919278e+00 |-7.607846e-01 |
|  209  | 5.2250 |  1   |   0.608336   |   0.568334   |  -4.813565   |-9.923281e+00 |-7.603606e-01 |
|  210  | 5.2500 |  1   |   0.608519   |   0.571894   |  -4.767231   |-9.927295e+00 |-7.599484e-01 |
|  211  | 5.2750 |  1   |   0.611513   |   0.573648   |  -4.664724   |-9.931315e+00 |-7.595504e-01 |
|  212  | 5.3000 |  1   |   0.663295   |   0.546204   |  -4.542805   |-9.935346e+00 |-7.591674e-01 |
|  213  | 5.3250 |  1   |   0.653387   |   0.540858   |  -4.498708   |-9.939397e+00 |-7.588020e-01 |
|  214  | 5.3500 |  1   |   0.651487   |   0.537849   |  -4.514320   |-9.943465e+00 |-7.584560e-01 |
|  215  | 5.3750 |  1   |   0.662014   |   0.545498   |  -4.531746   |-9.947549e+00 |-7.581246e-01 |
|  216  | 5.4000 |  1   |   0.650294   |   0.540917   |  -4.526842   |-9.951634e+00 |-7.578047e-01 |
|  217  | 5.4250 |  1   |   0.653085   |   0.536544   |  -4.480060   |-9.955710e+00 |-7.574922e-01 |
|  218  | 5.4500 |  1   |   0.661230   |   0.548448   |  -4.442806   |-9.959772e+00 |-7.571838e-01 |
|  219  | 5.4750 |  1   |   0.655229   |   0.543616   |  -4.442100   |-9.963823e+00 |-7.568782e-01 |
|  220  | 5.5000 |  1   |   0.655222   |   0.546932   |  -4.480309   |-9.967864e+00 |-7.565737e-01 |
|  221  | 5.5250 |  1   |   0.650105   |   0.542159   |  -4.527770   |-9.971886e+00 |-7.562652e-01 |
|  222  | 5.5500 |  1   |   0.651475   |   0.537507   |  -4.542362   |-9.975885e+00 |-7.559507e-01 |
|  223  | 5.5750 |  1   |   0.659401   |   0.547510   |  -4.506293   |-9.979850e+00 |-7.556271e-01 |
|  224  | 5.6000 |  1   |   0.601784   |   0.569276   |  -4.504088   |-9.983780e+00 |-7.552935e-01 |
|  225  | 5.6250 |  1   |   0.623897   |   0.557574   |  -4.512552   |-9.987687e+00 |-7.549518e-01 |
|  226  | 5.6500 |  1   |   0.600244   |   0.569273   |  -4.550006   |-9.991573e+00 |-7.546024e-01 |
|  227  | 5.6750 |  1   |   0.624355   |   0.556274   |  -4.608499   |-9.995440e+00 |-7.542473e-01 |
|  228  | 5.7000 |  1   |   0.660840   |   0.555257   |  -4.628774   |-9.999287e+00 |-7.538857e-01 |
|  229  | 5.7250 |  1   |   0.648993   |   0.540272   |  -4.638629   |-1.000311e+01 |-7.535194e-01 |
|  230  | 5.7500 |  1   |   0.649691   |   0.542594   |  -4.620373   |-1.000691e+01 |-7.531502e-01 |
|  231  | 5.7750 |  1   |   0.654206   |   0.542916   |  -4.623969   |-1.001071e+01 |-7.527814e-01 |
|  232  | 5.8000 |  1   |   0.651407   |   0.541567   |  -4.658962   |-1.001450e+01 |-7.524169e-01 |
|  233  | 5.8250 |  1   |   0.634053   |   0.546768   |  -4.709952   |-1.001828e+01 |-7.520571e-01 |
|  234  | 5.8500 |  1   |   0.649142   |   0.540074   |  -4.711982   |-1.002206e+01 |-7.517035e-01 |
|  235  | 5.8750 |  1   |   0.651331   |   0.541255   |  -4.674097   |-1.002584e+01 |-7.513553e-01 |
|  236  | 5.9000 |  1   |   0.650602   |   0.540557   |  -4.609681   |-1.002960e+01 |-7.510131e-01 |
|  237  | 5.9250 |  1   |   0.625181   |   0.556794   |  -4.572926   |-1.003336e+01 |-7.506790e-01 |
|  238  | 5.9500 |  1   |   0.598372   |   0.577537   |  -4.593342   |-1.003712e+01 |-7.503529e-01 |
|  239  | 5.9750 |  1   |   0.597025   |   0.572500   |  -4.606399   |-1.004088e+01 |-7.500343e-01 |
|  240  | 6.0000 |  1   |   0.597859   |   0.576480   |  -4.633256   |-1.004462e+01 |-7.497216e-01 |
|  241  | 6.0250 |  1   |   0.598875   |   0.573635   |  -4.596024   |-1.004836e+01 |-7.494121e-01 |
|  242  | 6.0500 |  1   |   0.597534   |   0.573711   |  -4.557101   |-1.005207e+01 |-7.491049e-01 |
|  243  | 6.0750 |  1   |   0.615834   |   0.560475   |  -4.544709   |-1.005577e+01 |-7.487993e-01 |
|  244  | 6.1000 |  1   |   0.645833   |   0.545940   |  -4.560556   |-1.005945e+01 |-7.484956e-01 |
|  245  | 6.1250 |  1   |   0.641262   |   0.550605   |  -4.599376   |-1.006312e+01 |-7.481921e-01 |
|  246  | 6.1500 |  1   |   0.650546   |   0.543172   |  -4.621850   |-1.006677e+01 |-7.478876e-01 |
|  247  | 6.1750 |  1   |   0.650686   |   0.547601   |  -4.613310   |-1.007040e+01 |-7.475803e-01 |
|  248  | 6.2000 |  1   |   0.643572   |   0.551210   |  -4.588594   |-1.007400e+01 |-7.472693e-01 |
|  249  | 6.2250 |  1   |   0.643333   |   0.539960   |  -4.585553   |-1.007758e+01 |-7.469564e-01 |
|  250  | 6.2500 |  1   |   0.631976   |   0.548090   |  -4.600869   |-1.008114e+01 |-7.466414e-01 |
|  251  | 6.2750 |  1   |   0.647855   |   0.541747   |  -4.650767   |-1.008469e+01 |-7.463261e-01 |
|  252  | 6.3000 |  1   |   0.649241   |   0.544522   |  -4.672520   |-1.008821e+01 |-7.460094e-01 |
|  253  | 6.3250 |  1   |   0.643516   |   0.545110   |  -4.673014   |-1.009172e+01 |-7.456916e-01 |
|  254  | 6.3500 |  1   |   0.638767   |   0.540989   |  -4.653989   |-1.009520e+01 |-7.453734e-01 |
|  255  | 6.3750 |  1   |   0.631475   |   0.546431   |  -4.631053   |-1.009867e+01 |-7.450557e-01 |
|  256  | 6.4000 |  1   |   0.644038   |   0.541411   |  -4.653231   |-1.010213e+01 |-7.447412e-01 |
|  257  | 6.4250 |  1   |   0.645015   |   0.541870   |  -4.678113   |-1.010557e+01 |-7.444295e-01 |
|  258  | 6.4500 |  1   |   0.646439   |   0.544957   |  -4.706046   |-1.010900e+01 |-7.441219e-01 |
|  259  | 6.4750 |  1   |   0.636618   |   0.544506   |  -4.688480   |-1.011242e+01 |-7.438168e-01 |
|  260  | 6.5000 |  1   |   0.636472   |   0.544515   |  -4.660864   |-1.011582e+01 |-7.435149e-01 |
|  261  | 6.5250 |  1   |   0.642801   |   0.547305   |  -4.622978   |-1.011920e+01 |-7.432168e-01 |
|  262  | 6.5500 |  1   |   0.642521   |   0.549237   |  -4.634656   |-1.012258e+01 |-7.429228e-01 |
|  263  | 6.5750 |  1   |   0.640326   |   0.546690   |  -4.652540   |-1.012594e+01 |-7.426333e-01 |
|  264  | 6.6000 |  1   |   0.646080   |   0.569616   |  -4.675082   |-1.012928e+01 |-7.423474e-01 |
|  265  | 6.6250 |  1   |   0.637068   |   0.545332   |  -4.671475   |-1.013261e+01 |-7.420636e-01 |
|  266  | 6.6500 |  1   |   0.641454   |   0.542948   |  -4.635937   |-1.013592e+01 |-7.417815e-01 |
|  267  | 6.6750 |  1   |   0.648429   |   0.563219   |  -4.626090   |-1.013921e+01 |-7.415010e-01 |
|  268  | 6.7000 |  1   |   0.638889   |   0.546413   |  -4.619397   |-1.014248e+01 |-7.412219e-01 |
|  269  | 6.7250 |  1   |   0.645476   |   0.551120   |  -4.654354   |-1.014573e+01 |-7.409449e-01 |
|  270  | 6.7500 |  1   |   0.644807   |   0.553462   |  -4.677256   |-1.014897e+01 |-7.406684e-01 |
|  271  | 6.7750 |  1   |   0.643342   |   0.545453   |  -4.677941   |-1.015219e+01 |-7.403922e-01 |
|  272  | 6.8000 |  1   |   0.642737   |   0.554577   |  -4.663328   |-1.015538e+01 |-7.401152e-01 |
|  273  | 6.8250 |  1   |   0.636924   |   0.547193   |  -4.647117   |-1.015854e+01 |-7.398382e-01 |
|  274  | 6.8500 |  1   |   0.636881   |   0.549125   |  -4.652275   |-1.016169e+01 |-7.395617e-01 |
|  275  | 6.8750 |  1   |   0.642097   |   0.546781   |  -4.679255   |-1.016482e+01 |-7.392860e-01 |
|  276  | 6.9000 |  1   |   0.644362   |   0.546495   |  -4.710093   |-1.016793e+01 |-7.390117e-01 |
|  277  | 6.9250 |  1   |   0.636971   |   0.552398   |  -4.709777   |-1.017103e+01 |-7.387375e-01 |
|  278  | 6.9500 |  1   |   0.642068   |   0.550708   |  -4.702989   |-1.017410e+01 |-7.384642e-01 |
|  279  | 6.9750 |  1   |   0.636245   |   0.548906   |  -4.675771   |-1.017714e+01 |-7.381918e-01 |
|  280  | 7.0000 |  1   |   0.631892   |   0.553548   |  -4.683037   |-1.018018e+01 |-7.379216e-01 |
|  281  | 7.0250 |  1   |   0.636876   |   0.548173   |  -4.700304   |-1.018319e+01 |-7.376540e-01 |
|  282  | 7.0500 |  1   |   0.641117   |   0.548960   |  -4.723800   |-1.018619e+01 |-7.373891e-01 |
|  283  | 7.0750 |  1   |   0.639323   |   0.547961   |  -4.733571   |-1.018917e+01 |-7.371265e-01 |
|  284  | 7.1000 |  1   |   0.636166   |   0.555813   |  -4.706688   |-1.019212e+01 |-7.368659e-01 |
|  285  | 7.1250 |  1   |   0.634238   |   0.555710   |  -4.691827   |-1.019506e+01 |-7.366075e-01 |
|  286  | 7.1500 |  1   |   0.635902   |   0.551450   |  -4.681559   |-1.019798e+01 |-7.363519e-01 |
|  287  | 7.1750 |  1   |   0.637927   |   0.565596   |  -4.702253   |-1.020088e+01 |-7.360992e-01 |
|  288  | 7.2000 |  1   |   0.636154   |   0.548763   |  -4.718294   |-1.020377e+01 |-7.358495e-01 |
|  289  | 7.2250 |  1   |   0.635841   |   0.549645   |  -4.734186   |-1.020663e+01 |-7.356017e-01 |
|  290  | 7.2500 |  1   |   0.633560   |   0.553457   |  -4.709277   |-1.020948e+01 |-7.353555e-01 |
|  291  | 7.2750 |  1   |   0.633063   |   0.551744   |  -4.697645   |-1.021230e+01 |-7.351108e-01 |
|  292  | 7.3000 |  1   |   0.596012   |   0.573846   |  -4.690652   |-1.021510e+01 |-7.348676e-01 |
|  293  | 7.3250 |  1   |   0.628225   |   0.550297   |  -4.705748   |-1.021788e+01 |-7.346267e-01 |
|  294  | 7.3500 |  1   |   0.635229   |   0.558756   |  -4.733700   |-1.022064e+01 |-7.343873e-01 |
|  295  | 7.3750 |  1   |   0.633064   |   0.553635   |  -4.738990   |-1.022338e+01 |-7.341492e-01 |
|  296  | 7.4000 |  1   |   0.636859   |   0.550407   |  -4.735953   |-1.022610e+01 |-7.339119e-01 |
|  297  | 7.4250 |  1   |   0.623950   |   0.565263   |  -4.716011   |-1.022879e+01 |-7.336753e-01 |
|  298  | 7.4500 |  1   |   0.642621   |   0.555373   |  -4.714487   |-1.023146e+01 |-7.334401e-01 |
|  299  | 7.4750 |  1   |   0.631974   |   0.548812   |  -4.730516   |-1.023411e+01 |-7.332063e-01 |
|  300  | 7.5000 |  1   |   0.635308   |   0.557555   |  -4.753499   |-1.023674e+01 |-7.329746e-01 |
|  301  | 7.5250 |  1   |   0.629950   |   0.550125   |  -4.765495   |-1.023935e+01 |-7.327439e-01 |
|  302  | 7.5500 |  1   |   0.615770   |   0.560338   |  -4.762381   |-1.024194e+01 |-7.325148e-01 |
|  303  | 7.5750 |  1   |   0.584256   |   0.590264   |  -4.742845   |-1.024450e+01 |-7.322869e-01 |
|  304  | 7.6000 |  1   |   0.582597   |   0.587076   |  -4.737265   |-1.024704e+01 |-7.320607e-01 |
|  305  | 7.6250 |  1   |   0.584283   |   0.579482   |  -4.752925   |-1.024956e+01 |-7.318370e-01 |
|  306  | 7.6500 |  1   |   0.586216   |   0.584941   |  -4.767222   |-1.025207e+01 |-7.316154e-01 |
|  307  | 7.6750 |  1   |   0.585430   |   0.579851   |  -4.787773   |-1.025455e+01 |-7.313962e-01 |
|  308  | 7.7000 |  1   |   0.631306   |   0.551397   |  -4.774572   |-1.025701e+01 |-7.311787e-01 |
|  309  | 7.7250 |  1   |   0.634317   |   0.553543   |  -4.762915   |-1.025944e+01 |-7.309630e-01 |
|  310  | 7.7500 |  1   |   0.636891   |   0.556693   |  -4.750823   |-1.026186e+01 |-7.307494e-01 |
|  311  | 7.7750 |  1   |   0.639769   |   0.563616   |  -4.761181   |-1.026425e+01 |-7.305383e-01 |
|  312  | 7.8000 |  1   |   0.631767   |   0.554912   |  -4.781589   |-1.026663e+01 |-7.303294e-01 |
|  313  | 7.8250 |  1   |   0.634194   |   0.558768   |  -4.793059   |-1.026898e+01 |-7.301229e-01 |
|  314  | 7.8500 |  1   |   0.626796   |   0.556363   |  -4.791256   |-1.027131e+01 |-7.299178e-01 |
|  315  | 7.8750 |  1   |   0.625787   |   0.550935   |  -4.775101   |-1.027361e+01 |-7.297145e-01 |
|  316  | 7.9000 |  1   |   0.631936   |   0.553434   |  -4.770127   |-1.027590e+01 |-7.295127e-01 |
|  317  | 7.9250 |  1   |   0.632807   |   0.554769   |  -4.771512   |-1.027816e+01 |-7.293129e-01 |
|  318  | 7.9500 |  1   |   0.636354   |   0.555106   |  -4.801316   |-1.028040e+01 |-7.291153e-01 |
|  319  | 7.9750 |  1   |   0.651982   |   0.601691   |  -4.806386   |-1.028262e+01 |-7.289192e-01 |
|  320  | 8.0000 |  1   |   0.642667   |   0.630753   |  -4.813430   |-1.028481e+01 |-7.287248e-01 |
|  321  | 8.0250 |  1   |   0.638641   |   0.626061   |  -4.798908   |-1.028698e+01 |-7.285314e-01 |
|  322  | 8.0500 |  1   |   0.627977   |   0.630209   |  -4.790992   |-1.028913e+01 |-7.283397e-01 |
|  323  | 8.0750 |  1   |   0.615535   |   0.632846   |  -4.799474   |-1.029125e+01 |-7.281498e-01 |
|  324  | 8.1000 |  1   |   0.628741   |   0.623770   |  -4.816641   |-1.029336e+01 |-7.279618e-01 |
|  325  | 8.1250 |  1   |   0.646119   |   0.631292   |  -4.835626   |-1.029544e+01 |-7.277758e-01 |
|  326  | 8.1500 |  1   |   0.635751   |   0.604086   |  -4.834189   |-1.029750e+01 |-7.275912e-01 |
|  327  | 8.1750 |  1   |   0.635591   |   0.596375   |  -4.826611   |-1.029953e+01 |-7.274083e-01 |
|  328  | 8.2000 |  1   |   0.654686   |   0.625780   |  -4.815149   |-1.030154e+01 |-7.272270e-01 |
|  329  | 8.2250 |  1   |   0.631972   |   0.593750   |  -4.822991   |-1.030353e+01 |-7.270479e-01 |
|  330  | 8.2500 |  1   |   0.627881   |   0.594475   |  -4.835602   |-1.030549e+01 |-7.268709e-01 |
|  331  | 8.2750 |  1   |   0.632779   |   0.588022   |  -4.856780   |-1.030744e+01 |-7.266960e-01 |
|  332  | 8.3000 |  1   |   0.623576   |   0.572123   |  -4.856935   |-1.030936e+01 |-7.265231e-01 |
|  333  | 8.3250 |  1   |   0.616926   |   0.572569   |  -4.844436   |-1.031125e+01 |-7.263517e-01 |
|  334  | 8.3500 |  1   |   0.656718   |   0.615642   |  -4.841857   |-1.031313e+01 |-7.261824e-01 |
|  335  | 8.3750 |  1   |   0.647848   |   0.593447   |  -4.836181   |-1.031498e+01 |-7.260150e-01 |
|  336  | 8.4000 |  1   |   0.620897   |   0.556400   |  -4.858940   |-1.031681e+01 |-7.258500e-01 |
|  337  | 8.4250 |  1   |   0.636878   |   0.568918   |  -4.870038   |-1.031861e+01 |-7.256870e-01 |
|  338  | 8.4500 |  1   |   0.624409   |   0.560472   |  -4.878751   |-1.032040e+01 |-7.255258e-01 |
|  339  | 8.4750 |  1   |   0.626291   |   0.560779   |  -4.866752   |-1.032216e+01 |-7.253663e-01 |
|  340  | 8.5000 |  1   |   0.624523   |   0.557208   |  -4.860873   |-1.032389e+01 |-7.252085e-01 |
|  341  | 8.5250 |  1   |   0.621932   |   0.557273   |  -4.861571   |-1.032560e+01 |-7.250526e-01 |
|  342  | 8.5500 |  1   |   0.620140   |   0.558312   |  -4.876236   |-1.032729e+01 |-7.248988e-01 |
|  343  | 8.5750 |  1   |   0.618488   |   0.558703   |  -4.894443   |-1.032896e+01 |-7.247469e-01 |
|  344  | 8.6000 |  1   |   0.614675   |   0.555611   |  -4.898885   |-1.033060e+01 |-7.245967e-01 |
|  345  | 8.6250 |  1   |   0.608933   |   0.564743   |  -4.897533   |-1.033222e+01 |-7.244482e-01 |
|  346  | 8.6500 |  1   |   0.577061   |   0.585133   |  -4.881841   |-1.033381e+01 |-7.243012e-01 |
|  347  | 8.6750 |  1   |   0.608953   |   0.587979   |  -4.892088   |-1.033539e+01 |-7.241562e-01 |
|  348  | 8.7000 |  1   |   0.624541   |   0.558283   |  -4.899079   |-1.033693e+01 |-7.240130e-01 |
|  349  | 8.7250 |  1   |   0.625952   |   0.557111   |  -4.918737   |-1.033846e+01 |-7.238720e-01 |
|  350  | 8.7500 |  1   |   0.623287   |   0.557909   |  -4.928123   |-1.033996e+01 |-7.237327e-01 |
|  351  | 8.7750 |  1   |   0.622417   |   0.555740   |  -4.921486   |-1.034144e+01 |-7.235950e-01 |
|  352  | 8.8000 |  1   |   0.621352   |   0.554043   |  -4.914257   |-1.034289e+01 |-7.234592e-01 |
|  353  | 8.8250 |  1   |   0.625442   |   0.559248   |  -4.914487   |-1.034432e+01 |-7.233251e-01 |
|  354  | 8.8500 |  1   |   0.621001   |   0.554895   |  -4.927093   |-1.034573e+01 |-7.231933e-01 |
|  355  | 8.8750 |  1   |   0.616331   |   0.556795   |  -4.942194   |-1.034711e+01 |-7.230634e-01 |
|  356  | 8.9000 |  1   |   0.620400   |   0.561975   |  -4.953444   |-1.034848e+01 |-7.229356e-01 |
|  357  | 8.9250 |  1   |   0.617789   |   0.557966   |  -4.948106   |-1.034981e+01 |-7.228093e-01 |
|  358  | 8.9500 |  1   |   0.617529   |   0.559993   |  -4.942308   |-1.035113e+01 |-7.226850e-01 |
|  359  | 8.9750 |  1   |   0.618381   |   0.557392   |  -4.938375   |-1.035241e+01 |-7.225625e-01 |
|  360  | 9.0000 |  1   |   0.615879   |   0.559019   |  -4.952078   |-1.035368e+01 |-7.224422e-01 |
|  361  | 9.0250 |  1   |   0.619261   |   0.558077   |  -4.968153   |-1.035492e+01 |-7.223238e-01 |
|  362  | 9.0500 |  1   |   0.620542   |   0.558342   |  -4.974721   |-1.035614e+01 |-7.222074e-01 |
|  363  | 9.0750 |  1   |   0.620811   |   0.562190   |  -4.979177   |-1.035734e+01 |-7.220928e-01 |
|  364  | 9.1000 |  1   |   0.621215   |   0.561517   |  -4.966813   |-1.035851e+01 |-7.219798e-01 |
|  365  | 9.1250 |  1   |   0.619293   |   0.556963   |  -4.968124   |-1.035966e+01 |-7.218687e-01 |
|  366  | 9.1500 |  1   |   0.617414   |   0.560206   |  -4.976628   |-1.036079e+01 |-7.217596e-01 |
|  367  | 9.1750 |  1   |   0.617224   |   0.560026   |  -4.995733   |-1.036189e+01 |-7.216525e-01 |
|  368  | 9.2000 |  1   |   0.594361   |   0.572718   |  -5.002079   |-1.036297e+01 |-7.215472e-01 |
|  369  | 9.2250 |  1   |   0.591752   |   0.620408   |  -5.007823   |-1.036402e+01 |-7.214437e-01 |
|  370  | 9.2500 |  1   |   0.573933   |   0.595363   |  -4.998087   |-1.036505e+01 |-7.213418e-01 |
|  371  | 9.2750 |  1   |   0.574780   |   0.599225   |  -4.997593   |-1.036606e+01 |-7.212418e-01 |
|  372  | 9.3000 |  1   |   0.573589   |   0.590403   |  -5.006758   |-1.036704e+01 |-7.211437e-01 |
|  373  | 9.3250 |  1   |   0.609827   |   0.562178   |  -5.021476   |-1.036800e+01 |-7.210476e-01 |
|  374  | 9.3500 |  1   |   0.614184   |   0.559384   |  -5.036294   |-1.036894e+01 |-7.209533e-01 |
|  375  | 9.3750 |  1   |   0.613003   |   0.559421   |  -5.033179   |-1.036985e+01 |-7.208608e-01 |
|  376  | 9.4000 |  1   |   0.615698   |   0.562315   |  -5.033218   |-1.037074e+01 |-7.207701e-01 |
|  377  | 9.4250 |  1   |   0.614765   |   0.561798   |  -5.027569   |-1.037161e+01 |-7.206811e-01 |
|  378  | 9.4500 |  1   |   0.617829   |   0.560943   |  -5.035541   |-1.037245e+01 |-7.205942e-01 |
|  379  | 9.4750 |  1   |   0.618720   |   0.564336   |  -5.051179   |-1.037327e+01 |-7.205092e-01 |
|  380  | 9.5000 |  1   |   0.612249   |   0.559879   |  -5.064730   |-1.037407e+01 |-7.204262e-01 |
|  381  | 9.5250 |  1   |   0.616751   |   0.558819   |  -5.064004   |-1.037484e+01 |-7.203449e-01 |
|  382  | 9.5500 |  1   |   0.609852   |   0.562424   |  -5.062977   |-1.037559e+01 |-7.202654e-01 |
|  383  | 9.5750 |  1   |   0.610932   |   0.561283   |  -5.059836   |-1.037632e+01 |-7.201877e-01 |
|  384  | 9.6000 |  1   |   0.609683   |   0.559666   |  -5.063394   |-1.037702e+01 |-7.201119e-01 |
|  385  | 9.6250 |  1   |   0.614492   |   0.563530   |  -5.081452   |-1.037770e+01 |-7.200382e-01 |
|  386  | 9.6500 |  1   |   0.606921   |   0.560035   |  -5.090793   |-1.037835e+01 |-7.199663e-01 |
|  387  | 9.6750 |  1   |   0.614644   |   0.567031   |  -5.097605   |-1.037899e+01 |-7.198962e-01 |
|  388  | 9.7000 |  1   |   0.584276   |   0.590999   |  -5.091026   |-1.037960e+01 |-7.198278e-01 |
|  389  | 9.7250 |  1   |   0.567643   |   0.593811   |  -5.091591   |-1.038018e+01 |-7.197613e-01 |
|  390  | 9.7500 |  1   |   0.567374   |   0.596893   |  -5.095862   |-1.038075e+01 |-7.196965e-01 |
|  391  | 9.7750 |  1   |   0.572616   |   0.596802   |  -5.106980   |-1.038129e+01 |-7.196338e-01 |
|  392  | 9.8000 |  1   |   0.570078   |   0.591577   |  -5.123029   |-1.038181e+01 |-7.195729e-01 |
|  393  | 9.8250 |  1   |   0.568206   |   0.592200   |  -5.126210   |-1.038230e+01 |-7.195138e-01 |
|  394  | 9.8500 |  1   |   0.571314   |   0.611958   |  -5.123923   |-1.038277e+01 |-7.194564e-01 |
|  395  | 9.8750 |  1   |   0.567253   |   0.597897   |  -5.121337   |-1.038322e+01 |-7.194006e-01 |
|  396  | 9.9000 |  1   |   0.568210   |   0.593683   |  -5.128361   |-1.038365e+01 |-7.193469e-01 |
|  397  | 9.9250 |  1   |   0.565010   |   0.597851   |  -5.135472   |-1.038405e+01 |-7.192949e-01 |
|  398  | 9.9500 |  1   |   0.607473   |   0.563262   |  -5.151271   |-1.038443e+01 |-7.192449e-01 |
|  399  | 9.9750 |  1   |   0.611926   |   0.565768   |  -5.156484   |-1.038479e+01 |-7.191966e-01 |
|  400  |10.0000 |  1   |   0.609614   |   0.566774   |  -5.154040   |-1.038512e+01 |-7.191501e-01 |
|  401  |10.0250 |  1   |   0.609190   |   0.563351   |  -5.151962   |-1.038543e+01 |-7.191053e-01 |
|  402  |10.0500 |  1   |   0.609842   |   0.560413   |  -5.156639   |-1.038572e+01 |-7.190623e-01 |
|  403  |10.0750 |  1   |   0.608545   |   0.562862   |  -5.165364   |-1.038599e+01 |-7.190212e-01 |
|  404  |10.1000 |  1   |   0.609280   |   0.574591   |  -5.175709   |-1.038623e+01 |-7.189820e-01 |
|  405  |10.1250 |  1   |   0.606715   |   0.563175   |  -5.184143   |-1.038645e+01 |-7.189446e-01 |
|  406  |10.1500 |  1   |   0.604479   |   0.564748   |  -5.181991   |-1.038665e+01 |-7.189088e-01 |
|  407  |10.1750 |  1   |   0.606050   |   0.565112   |  -5.179552   |-1.038683e+01 |-7.188749e-01 |
|  408  |10.2000 |  1   |   0.607161   |   0.564970   |  -5.182516   |-1.038698e+01 |-7.188427e-01 |
|  409  |10.2250 |  1   |   0.611589   |   0.565026   |  -5.191318   |-1.038711e+01 |-7.188124e-01 |
|  410  |10.2500 |  1   |   0.609824   |   0.564363   |  -5.199527   |-1.038722e+01 |-7.187839e-01 |
|  411  |10.2750 |  1   |   0.608612   |   0.565428   |  -5.204957   |-1.038731e+01 |-7.187572e-01 |
|  412  |10.3000 |  1   |   0.607257   |   0.561791   |  -5.208673   |-1.038738e+01 |-7.187322e-01 |
|  413  |10.3250 |  1   |   0.609691   |   0.568279   |  -5.201979   |-1.038742e+01 |-7.187089e-01 |
|  414  |10.3500 |  1   |   0.601743   |   0.568179   |  -5.206076   |-1.038744e+01 |-7.186873e-01 |
|  415  |10.3750 |  1   |   0.601717   |   0.567023   |  -5.212946   |-1.038744e+01 |-7.186677e-01 |
|  416  |10.4000 |  1   |   0.559912   |   0.595790   |  -5.218419   |-1.038742e+01 |-7.186497e-01 |
|  417  |10.4250 |  1   |   0.564230   |   0.598142   |  -5.224424   |-1.038737e+01 |-7.186336e-01 |
|  418  |10.4500 |  1   |   0.583181   |   0.581345   |  -5.225489   |-1.038730e+01 |-7.186191e-01 |
|  419  |10.4750 |  1   |   0.602301   |   0.565017   |  -5.224159   |-1.038721e+01 |-7.186063e-01 |
|  420  |10.5000 |  1   |   0.603593   |   0.566494   |  -5.222729   |-1.038710e+01 |-7.185953e-01 |
|  421  |10.5250 |  1   |   0.601339   |   0.567735   |  -5.228588   |-1.038697e+01 |-7.185860e-01 |
|  422  |10.5500 |  1   |   0.600440   |   0.570387   |  -5.234701   |-1.038682e+01 |-7.185785e-01 |
|  423  |10.5750 |  1   |   0.578888   |   0.583202   |  -5.234590   |-1.038664e+01 |-7.185728e-01 |
|  424  |10.6000 |  1   |   0.604815   |   0.565890   |  -5.239249   |-1.038645e+01 |-7.185687e-01 |
|  425  |10.6250 |  1   |   0.604126   |   0.569152   |  -5.236159   |-1.038623e+01 |-7.185663e-01 |
|  426  |10.6500 |  1   |   0.603547   |   0.566174   |  -5.235546   |-1.038599e+01 |-7.185655e-01 |
|  427  |10.6750 |  1   |   0.602865   |   0.569464   |  -5.237490   |-1.038573e+01 |-7.185666e-01 |
|  428  |10.7000 |  1   |   0.600635   |   0.568652   |  -5.241270   |-1.038545e+01 |-7.185694e-01 |
|  429  |10.7250 |  1   |   0.605226   |   0.574795   |  -5.242437   |-1.038514e+01 |-7.185739e-01 |
|  430  |10.7500 |  1   |   0.600788   |   0.571143   |  -5.241394   |-1.038482e+01 |-7.185801e-01 |
|  431  |10.7750 |  1   |   0.577222   |   0.591226   |  -5.242569   |-1.038448e+01 |-7.185880e-01 |
|  432  |10.8000 |  1   |   0.560620   |   0.597055   |  -5.240958   |-1.038411e+01 |-7.185975e-01 |
|  433  |10.8250 |  1   |   0.556709   |   0.596082   |  -5.238278   |-1.038372e+01 |-7.186087e-01 |
|  434  |10.8500 |  1   |   0.556684   |   0.597562   |  -5.242316   |-1.038332e+01 |-7.186217e-01 |
|  435  |10.8750 |  1   |   0.559204   |   0.599864   |  -5.240932   |-1.038289e+01 |-7.186363e-01 |
|  436  |10.9000 |  1   |   0.554305   |   0.620321   |  -5.238040   |-1.038244e+01 |-7.186526e-01 |
|  437  |10.9250 |  1   |   0.577513   |   0.588008   |  -5.241438   |-1.038197e+01 |-7.186705e-01 |
|  438  |10.9500 |  1   |   0.599971   |   0.568971   |  -5.235789   |-1.038148e+01 |-7.186901e-01 |
|  439  |10.9750 |  1   |   0.599382   |   0.571610   |  -5.237054   |-1.038097e+01 |-7.187113e-01 |
|  440  |11.0000 |  1   |   0.597518   |   0.569614   |  -5.232763   |-1.038044e+01 |-7.187342e-01 |
|  441  |11.0250 |  1   |   0.603696   |   0.575074   |  -5.233276   |-1.037989e+01 |-7.187588e-01 |
|  442  |11.0500 |  1   |   0.583094   |   0.697022   |  -5.231065   |-1.037932e+01 |-7.187850e-01 |
|  443  |11.0750 |  1   |   0.619875   |   0.676204   |  -5.227894   |-1.037873e+01 |-7.188128e-01 |
|  444  |11.1000 |  1   |   0.598458   |   0.604789   |  -5.230200   |-1.037812e+01 |-7.188422e-01 |
|  445  |11.1250 |  1   |   0.604274   |   0.596907   |  -5.223329   |-1.037749e+01 |-7.188732e-01 |
|  446  |11.1500 |  1   |   0.573906   |   0.610394   |  -5.222229   |-1.037684e+01 |-7.189059e-01 |
|  447  |11.1750 |  1   |   0.561556   |   0.624831   |  -5.217935   |-1.037617e+01 |-7.189403e-01 |
|  448  |11.2000 |  1   |   0.569148   |   0.626453   |  -5.216896   |-1.037548e+01 |-7.189761e-01 |
|  449  |11.2250 |  1   |   0.564965   |   0.673984   |  -5.213489   |-1.037477e+01 |-7.190136e-01 |
|  450  |11.2500 |  1   |   0.561458   |   0.614712   |  -5.212846   |-1.037404e+01 |-7.190527e-01 |
|  451  |11.2750 |  1   |   0.573335   |   0.635211   |  -5.209958   |-1.037329e+01 |-7.190933e-01 |
|  452  |11.3000 |  1   |   0.601036   |   0.589727   |  -5.203174   |-1.037253e+01 |-7.191356e-01 |
|  453  |11.3250 |  1   |   0.588650   |   0.649352   |  -5.201050   |-1.037174e+01 |-7.191795e-01 |
|  454  |11.3500 |  1   |   0.581506   |   0.609257   |  -5.196793   |-1.037094e+01 |-7.192250e-01 |
|  455  |11.3750 |  1   |   0.589146   |   0.602935   |  -5.195584   |-1.037011e+01 |-7.192719e-01 |
|  456  |11.4000 |  1   |   0.604540   |   0.602902   |  -5.192228   |-1.036927e+01 |-7.193204e-01 |
|  457  |11.4250 |  1   |   0.600579   |   0.594754   |  -5.191012   |-1.036841e+01 |-7.193705e-01 |
|  458  |11.4500 |  1   |   0.594932   |   0.585988   |  -5.182644   |-1.036753e+01 |-7.194222e-01 |
|  459  |11.4750 |  1   |   0.594136   |   0.587465   |  -5.180474   |-1.036663e+01 |-7.194754e-01 |
|  460  |11.5000 |  1   |   0.615072   |   0.617353   |  -5.174101   |-1.036571e+01 |-7.195302e-01 |
|  461  |11.5250 |  1   |   0.580867   |   0.632210   |  -5.174853   |-1.036478e+01 |-7.195865e-01 |
|  462  |11.5500 |  1   |   0.594978   |   0.646045   |  -5.170050   |-1.036382e+01 |-7.196442e-01 |
|  463  |11.5750 |  1   |   0.578537   |   0.617545   |  -5.167418   |-1.036285e+01 |-7.197035e-01 |
|  464  |11.6000 |  1   |   0.591423   |   0.656918   |  -5.163354   |-1.036186e+01 |-7.197643e-01 |
|  465  |11.6250 |  1   |   0.596206   |   0.593319   |  -5.154243   |-1.036085e+01 |-7.198267e-01 |
|  466  |11.6500 |  1   |   0.594452   |   0.615504   |  -5.154289   |-1.035983e+01 |-7.198905e-01 |
|  467  |11.6750 |  1   |   0.585962   |   0.589556   |  -5.148727   |-1.035878e+01 |-7.199559e-01 |
|  468  |11.7000 |  1   |   0.590704   |   0.592317   |  -5.148438   |-1.035772e+01 |-7.200226e-01 |
|  469  |11.7250 |  1   |   0.588268   |   0.610512   |  -5.143636   |-1.035664e+01 |-7.200909e-01 |
|  470  |11.7500 |  1   |   0.613000   |   0.604874   |  -5.138982   |-1.035554e+01 |-7.201607e-01 |
|  471  |11.7750 |  1   |   0.597844   |   0.624120   |  -5.133073   |-1.035443e+01 |-7.202319e-01 |
|  472  |11.8000 |  1   |   0.612249   |   0.607180   |  -5.128111   |-1.035330e+01 |-7.203046e-01 |
|  473  |11.8250 |  1   |   0.589453   |   0.613552   |  -5.126736   |-1.035215e+01 |-7.203787e-01 |
|  474  |11.8500 |  1   |   0.593775   |   0.611720   |  -5.123366   |-1.035098e+01 |-7.204543e-01 |
|  475  |11.8750 |  1   |   0.589657   |   0.587944   |  -5.120797   |-1.034980e+01 |-7.205313e-01 |
|  476  |11.9000 |  1   |   0.589097   |   0.606891   |  -5.114568   |-1.034860e+01 |-7.206098e-01 |
|  477  |11.9250 |  1   |   0.587495   |   0.629681   |  -5.110620   |-1.034739e+01 |-7.206897e-01 |
|  478  |11.9500 |  1   |   0.584468   |   0.608920   |  -5.103850   |-1.034615e+01 |-7.207710e-01 |
|  479  |11.9750 |  1   |   0.590653   |   0.613406   |  -5.103272   |-1.034491e+01 |-7.208537e-01 |
|  480  |12.0000 |  1   |   0.622715   |   0.666483   |  -5.099819   |-1.034364e+01 |-7.209378e-01 |
|  481  |12.0250 |  1   |   0.604598   |   0.600735   |  -5.096876   |-1.034236e+01 |-7.210233e-01 |
|  482  |12.0500 |  1   |   0.607466   |   0.600419   |  -5.092345   |-1.034106e+01 |-7.211102e-01 |
|  483  |12.0750 |  1   |   0.611803   |   0.651007   |  -5.086132   |-1.033975e+01 |-7.211985e-01 |
|  484  |12.1000 |  1   |   0.600163   |   0.604296   |  -5.082930   |-1.033842e+01 |-7.212882e-01 |
|  485  |12.1250 |  1   |   0.597081   |   0.608633   |  -5.078037   |-1.033707e+01 |-7.213792e-01 |
|  486  |12.1500 |  1   |   0.599493   |   0.694460   |  -5.078534   |-1.033571e+01 |-7.214716e-01 |
|  487  |12.1750 |  1   |   0.569671   |   0.611804   |  -5.072894   |-1.033433e+01 |-7.215653e-01 |
|  488  |12.2000 |  1   |   0.590231   |   0.592073   |  -5.070548   |-1.033294e+01 |-7.216604e-01 |
|  489  |12.2250 |  1   |   0.586426   |   0.587308   |  -5.063791   |-1.033153e+01 |-7.217569e-01 |
|  490  |12.2500 |  1   |   0.605616   |   0.639888   |  -5.060453   |-1.033011e+01 |-7.218546e-01 |
|  491  |12.2750 |  1   |   0.601391   |   0.589785   |  -5.056911   |-1.032867e+01 |-7.219537e-01 |
|  492  |12.3000 |  1   |   0.583376   |   0.588843   |  -5.054795   |-1.032722e+01 |-7.220541e-01 |
|  493  |12.3250 |  1   |   0.591293   |   0.608539   |  -5.052808   |-1.032575e+01 |-7.221558e-01 |
|  494  |12.3500 |  1   |   0.568992   |   0.595768   |  -5.047491   |-1.032427e+01 |-7.222588e-01 |
|  495  |12.3750 |  1   |   0.563630   |   0.599818   |  -5.043728   |-1.032277e+01 |-7.223631e-01 |
|  496  |12.4000 |  1   |   0.583918   |   0.582994   |  -5.038711   |-1.032126e+01 |-7.224687e-01 |
|  497  |12.4250 |  1   |   0.587738   |   0.584071   |  -5.036205   |-1.031973e+01 |-7.225755e-01 |
|  498  |12.4500 |  1   |   0.575600   |   0.588839   |  -5.033715   |-1.031819e+01 |-7.226837e-01 |
|  499  |12.4750 |  1   |   0.586772   |   0.589606   |  -5.031815   |-1.031664e+01 |-7.227930e-01 |
|  500  |12.5000 |  1   |   0.577898   |   0.602334   |  -5.027416   |-1.031507e+01 |-7.229037e-01 |
|  501  |12.5250 |  1   |   0.585324   |   0.589641   |  -5.023384   |-1.031348e+01 |-7.230155e-01 |
|  502  |12.5500 |  1   |   0.585512   |   0.584594   |  -5.019042   |-1.031189e+01 |-7.231287e-01 |
|  503  |12.5750 |  1   |   0.583865   |   0.582880   |  -5.016063   |-1.031028e+01 |-7.232430e-01 |
|  504  |12.6000 |  1   |   0.588496   |   0.592948   |  -5.014501   |-1.030865e+01 |-7.233586e-01 |
|  505  |12.6250 |  1   |   0.587081   |   0.593772   |  -5.011221   |-1.030702e+01 |-7.234753e-01 |
|  506  |12.6500 |  1   |   0.585744   |   0.587407   |  -5.009390   |-1.030537e+01 |-7.235933e-01 |
|  507  |12.6750 |  1   |   0.584881   |   0.596360   |  -5.003415   |-1.030370e+01 |-7.237125e-01 |
|  508  |12.7000 |  1   |   0.596729   |   0.591525   |  -5.001095   |-1.030203e+01 |-7.238329e-01 |
|  509  |12.7250 |  1   |   0.596917   |   0.597681   |  -4.997208   |-1.030034e+01 |-7.239544e-01 |
|  510  |12.7500 |  1   |   0.591709   |   0.586425   |  -4.995682   |-1.029863e+01 |-7.240771e-01 |
|  511  |12.7750 |  1   |   0.587641   |   0.591100   |  -4.993524   |-1.029692e+01 |-7.242010e-01 |
|  512  |12.8000 |  1   |   0.587204   |   0.586659   |  -4.990360   |-1.029519e+01 |-7.243260e-01 |
|  513  |12.8250 |  1   |   0.591092   |   0.587204   |  -4.986833   |-1.029345e+01 |-7.244521e-01 |
|  514  |12.8500 |  1   |   0.585463   |   0.583301   |  -4.982664   |-1.029170e+01 |-7.245795e-01 |
|  515  |12.8750 |  1   |   0.589045   |   0.586801   |  -4.980566   |-1.028993e+01 |-7.247079e-01 |
|  516  |12.9000 |  1   |   0.596714   |   0.601805   |  -4.977829   |-1.028816e+01 |-7.248374e-01 |
|  517  |12.9250 |  1   |   0.584486   |   0.594004   |  -4.976834   |-1.028637e+01 |-7.249680e-01 |
|  518  |12.9500 |  1   |   0.577005   |   0.592874   |  -4.973053   |-1.028457e+01 |-7.250998e-01 |
|  519  |12.9750 |  1   |   0.576873   |   0.590686   |  -4.970798   |-1.028276e+01 |-7.252326e-01 |
|  520  |13.0000 |  1   |   0.588123   |   0.584250   |  -4.966119   |-1.028094e+01 |-7.253666e-01 |
|  521  |13.0250 |  1   |   0.588080   |   0.592052   |  -4.964473   |-1.027910e+01 |-7.255016e-01 |
|  522  |13.0500 |  1   |   0.586211   |   0.588701   |  -4.961953   |-1.027726e+01 |-7.256376e-01 |
|  523  |13.0750 |  1   |   0.582357   |   0.595669   |  -4.960468   |-1.027540e+01 |-7.257747e-01 |
|  524  |13.1000 |  1   |   0.569600   |   0.594107   |  -4.958069   |-1.027353e+01 |-7.259128e-01 |
|  525  |13.1250 |  1   |   0.575376   |   0.594160   |  -4.954661   |-1.027165e+01 |-7.260521e-01 |
|  526  |13.1500 |  1   |   0.578324   |   0.588696   |  -4.952012   |-1.026976e+01 |-7.261923e-01 |
|  527  |13.1750 |  1   |   0.587095   |   0.592266   |  -4.948571   |-1.026786e+01 |-7.263335e-01 |
|  528  |13.2000 |  1   |   0.569050   |   0.599524   |  -4.947926   |-1.026595e+01 |-7.264758e-01 |
|  529  |13.2250 |  1   |   0.542086   |   0.614471   |  -4.945114   |-1.026403e+01 |-7.266190e-01 |
|  530  |13.2500 |  1   |   0.540521   |   0.611883   |  -4.943929   |-1.026210e+01 |-7.267632e-01 |
|  531  |13.2750 |  1   |   0.543265   |   0.621184   |  -4.940566   |-1.026016e+01 |-7.269084e-01 |
|  532  |13.3000 |  1   |   0.541681   |   0.615142   |  -4.937617   |-1.025821e+01 |-7.270546e-01 |
|  533  |13.3250 |  1   |   0.540352   |   0.616832   |  -4.935709   |-1.025625e+01 |-7.272018e-01 |
|  534  |13.3500 |  1   |   0.544213   |   0.620769   |  -4.933038   |-1.025428e+01 |-7.273499e-01 |
|  535  |13.3750 |  1   |   0.580695   |   0.590777   |  -4.932776   |-1.025230e+01 |-7.274989e-01 |
|  536  |13.4000 |  1   |   0.575465   |   0.596223   |  -4.929595   |-1.025031e+01 |-7.276489e-01 |
|  537  |13.4250 |  1   |   0.542536   |   0.618744   |  -4.928176   |-1.024831e+01 |-7.277998e-01 |
|  538  |13.4500 |  1   |   0.580911   |   0.591514   |  -4.924586   |-1.024631e+01 |-7.279516e-01 |
|  539  |13.4750 |  1   |   0.581108   |   0.591034   |  -4.922988   |-1.024429e+01 |-7.281044e-01 |
|  540  |13.5000 |  1   |   0.545993   |   0.614974   |  -4.920767   |-1.024227e+01 |-7.282580e-01 |
|  541  |13.5250 |  1   |   0.545921   |   0.615012   |  -4.919915   |-1.024023e+01 |-7.284125e-01 |
|  542  |13.5500 |  1   |   0.542495   |   0.613665   |  -4.917870   |-1.023819e+01 |-7.285679e-01 |
|  543  |13.5750 |  1   |   0.536964   |   0.622532   |  -4.915675   |-1.023614e+01 |-7.287241e-01 |
|  544  |13.6000 |  1   |   0.554106   |   0.603419   |  -4.913529   |-1.023408e+01 |-7.288812e-01 |
|  545  |13.6250 |  1   |   0.578840   |   0.604562   |  -4.910680   |-1.023202e+01 |-7.290392e-01 |
|  546  |13.6500 |  1   |   0.585058   |   0.606280   |  -4.910137   |-1.022994e+01 |-7.291979e-01 |
|  547  |13.6750 |  1   |   0.579318   |   0.591994   |  -4.907749   |-1.022786e+01 |-7.293575e-01 |
|  548  |13.7000 |  1   |   0.580380   |   0.589279   |  -4.907371   |-1.022577e+01 |-7.295179e-01 |
|  549  |13.7250 |  1   |   0.584794   |   0.591958   |  -4.904445   |-1.022367e+01 |-7.296792e-01 |
|  550  |13.7500 |  1   |   0.588687   |   0.601052   |  -4.902856   |-1.022157e+01 |-7.298412e-01 |
|  551  |13.7750 |  1   |   0.579102   |   0.595429   |  -4.900526   |-1.021945e+01 |-7.300040e-01 |
|  552  |13.8000 |  1   |   0.579922   |   0.596135   |  -4.899156   |-1.021733e+01 |-7.301676e-01 |
|  553  |13.8250 |  1   |   0.579598   |   0.597193   |  -4.898152   |-1.021521e+01 |-7.303320e-01 |
|  554  |13.8500 |  1   |   0.589419   |   0.608047   |  -4.896538   |-1.021307e+01 |-7.304971e-01 |
|  555  |13.8750 |  1   |   0.584321   |   0.601955   |  -4.895262   |-1.021093e+01 |-7.306629e-01 |
|  556  |13.9000 |  1   |   0.579038   |   0.593430   |  -4.892631   |-1.020879e+01 |-7.308295e-01 |
|  557  |13.9250 |  1   |   0.577858   |   0.591474   |  -4.891491   |-1.020663e+01 |-7.309968e-01 |
|  558  |13.9500 |  1   |   0.578370   |   0.601104   |  -4.889505   |-1.020447e+01 |-7.311649e-01 |
|  559  |13.9750 |  1   |   0.574518   |   0.594961   |  -4.888943   |-1.020231e+01 |-7.313336e-01 |
|  560  |14.0000 |  1   |   0.578717   |   0.588370   |  -4.887586   |-1.020014e+01 |-7.315030e-01 |
|  561  |14.0250 |  1   |   0.569221   |   0.592313   |  -4.886037   |-1.019796e+01 |-7.316731e-01 |
|  562  |14.0500 |  1   |   0.569323   |   0.588773   |  -4.884458   |-1.019577e+01 |-7.318439e-01 |
|  563  |14.0750 |  1   |   0.575350   |   0.590234   |  -4.882463   |-1.019359e+01 |-7.320154e-01 |
|  564  |14.1000 |  1   |   0.565140   |   0.597502   |  -4.881607   |-1.019139e+01 |-7.321875e-01 |
|  565  |14.1250 |  1   |   0.555995   |   0.601872   |  -4.880250   |-1.018919e+01 |-7.323602e-01 |
|  566  |14.1500 |  1   |   0.582065   |   0.599824   |  -4.879771   |-1.018699e+01 |-7.325336e-01 |
|  567  |14.1750 |  1   |   0.573287   |   0.591480   |  -4.877873   |-1.018478e+01 |-7.327076e-01 |
|  568  |14.2000 |  1   |   0.574334   |   0.602253   |  -4.876963   |-1.018256e+01 |-7.328822e-01 |
|  569  |14.2250 |  1   |   0.531243   |   0.628437   |  -4.874752   |-1.018034e+01 |-7.330574e-01 |
|  570  |14.2500 |  1   |   0.530046   |   0.617958   |  -4.874261   |-1.017812e+01 |-7.332332e-01 |
|  571  |14.2750 |  1   |   0.533493   |   0.621578   |  -4.872975   |-1.017589e+01 |-7.334095e-01 |
|  572  |14.3000 |  1   |   0.531280   |   0.620959   |  -4.872344   |-1.017365e+01 |-7.335865e-01 |
|  573  |14.3250 |  1   |   0.533888   |   0.617618   |  -4.871330   |-1.017142e+01 |-7.337640e-01 |
|  574  |14.3500 |  1   |   0.538329   |   0.615710   |  -4.869608   |-1.016917e+01 |-7.339420e-01 |
|  575  |14.3750 |  1   |   0.538880   |   0.617203   |  -4.868750   |-1.016693e+01 |-7.341206e-01 |
|  576  |14.4000 |  1   |   0.538717   |   0.619451   |  -4.867217   |-1.016468e+01 |-7.342997e-01 |
|  577  |14.4250 |  1   |   0.521259   |   0.628772   |  -4.866965   |-1.016243e+01 |-7.344794e-01 |
|  578  |14.4500 |  1   |   0.532905   |   0.622927   |  -4.865877   |-1.016017e+01 |-7.346595e-01 |
|  579  |14.4750 |  1   |   0.533617   |   0.622496   |  -4.865300   |-1.015791e+01 |-7.348401e-01 |
|  580  |14.5000 |  1   |   0.534588   |   0.619178   |  -4.863812   |-1.015565e+01 |-7.350212e-01 |
|  581  |14.5250 |  1   |   0.541063   |   0.619951   |  -4.862879   |-1.015338e+01 |-7.352028e-01 |
|  582  |14.5500 |  1   |   0.552002   |   0.630617   |  -4.861832   |-1.015111e+01 |-7.353848e-01 |
|  583  |14.5750 |  1   |   0.534411   |   0.618651   |  -4.861210   |-1.014884e+01 |-7.355673e-01 |
|  584  |14.6000 |  1   |   0.519947   |   0.634783   |  -4.860831   |-1.014656e+01 |-7.357502e-01 |
|  585  |14.6250 |  1   |   0.532511   |   0.623628   |  -4.859751   |-1.014429e+01 |-7.359336e-01 |
|  586  |14.6500 |  1   |   0.533837   |   0.631545   |  -4.859199   |-1.014201e+01 |-7.361174e-01 |
|  587  |14.6750 |  1   |   0.531337   |   0.621002   |  -4.857699   |-1.013972e+01 |-7.363016e-01 |
|  588  |14.7000 |  1   |   0.533695   |   0.628152   |  -4.857362   |-1.013744e+01 |-7.364862e-01 |
|  589  |14.7250 |  1   |   0.537053   |   0.620282   |  -4.856500   |-1.013515e+01 |-7.366711e-01 |
|  590  |14.7500 |  1   |   0.530726   |   0.624015   |  -4.856289   |-1.013286e+01 |-7.368564e-01 |
|  591  |14.7750 |  1   |   0.536236   |   0.627392   |  -4.855632   |-1.013057e+01 |-7.370421e-01 |
|  592  |14.8000 |  1   |   0.543870   |   0.627993   |  -4.854757   |-1.012828e+01 |-7.372282e-01 |
|  593  |14.8250 |  1   |   0.526978   |   0.622442   |  -4.854112   |-1.012599e+01 |-7.374146e-01 |
|  594  |14.8500 |  1   |   0.533051   |   0.620095   |  -4.853078   |-1.012369e+01 |-7.376013e-01 |
|  595  |14.8750 |  1   |   0.537832   |   0.629321   |  -4.853194   |-1.012140e+01 |-7.377884e-01 |
|  596  |14.9000 |  1   |   0.534586   |   0.618118   |  -4.852261   |-1.011910e+01 |-7.379757e-01 |
|  597  |14.9250 |  1   |   0.567604   |   0.593661   |  -4.852487   |-1.011680e+01 |-7.381634e-01 |
|  598  |14.9500 |  1   |   0.572954   |   0.599532   |  -4.851275   |-1.011451e+01 |-7.383513e-01 |
|  599  |14.9750 |  1   |   0.562174   |   0.595357   |  -4.851012   |-1.011221e+01 |-7.385395e-01 |
|  600  |15.0000 |  1   |   0.565041   |   0.594815   |  -4.850216   |-1.010991e+01 |-7.387280e-01 |
|  601  |15.0250 |  1   |   0.568619   |   0.602458   |  -4.849993   |-1.010761e+01 |-7.389167e-01 |
|  602  |15.0500 |  1   |   0.565685   |   0.599703   |  -4.849887   |-1.010531e+01 |-7.391057e-01 |
|  603  |15.0750 |  1   |   0.559376   |   0.612765   |  -4.849425   |-1.010300e+01 |-7.392949e-01 |
|  604  |15.1000 |  1   |   0.558552   |   0.602107   |  -4.849206   |-1.010070e+01 |-7.394843e-01 |
|  605  |15.1250 |  1   |   0.582718   |   0.633646   |  -4.848406   |-1.009840e+01 |-7.396740e-01 |
|  606  |15.1500 |  1   |   0.581577   |   0.623610   |  -4.848264   |-1.009610e+01 |-7.398638e-01 |
|  607  |15.1750 |  1   |   0.565744   |   0.594525   |  -4.847774   |-1.009380e+01 |-7.400538e-01 |
|  608  |15.2000 |  1   |   0.564061   |   0.594557   |  -4.848037   |-1.009150e+01 |-7.402440e-01 |
|  609  |15.2250 |  1   |   0.567888   |   0.599122   |  -4.847578   |-1.008921e+01 |-7.404344e-01 |
|  610  |15.2500 |  1   |   0.570234   |   0.602823   |  -4.847542   |-1.008691e+01 |-7.406249e-01 |
|  611  |15.2750 |  1   |   0.561263   |   0.598452   |  -4.847030   |-1.008461e+01 |-7.408156e-01 |
|  612  |15.3000 |  1   |   0.568963   |   0.599018   |  -4.846710   |-1.008231e+01 |-7.410064e-01 |
|  613  |15.3250 |  1   |   0.559163   |   0.605724   |  -4.846813   |-1.008002e+01 |-7.411973e-01 |
|  614  |15.3500 |  1   |   0.564462   |   0.599463   |  -4.846562   |-1.007773e+01 |-7.413883e-01 |
|  615  |15.3750 |  1   |   0.580952   |   0.610074   |  -4.846943   |-1.007543e+01 |-7.415794e-01 |
|  616  |15.4000 |  1   |   0.568574   |   0.597096   |  -4.846413   |-1.007314e+01 |-7.417706e-01 |
|  617  |15.4250 |  1   |   0.565757   |   0.596118   |  -4.846541   |-1.007085e+01 |-7.419619e-01 |
|  618  |15.4500 |  1   |   0.537671   |   0.632581   |  -4.846117   |-1.006857e+01 |-7.421533e-01 |
|  619  |15.4750 |  1   |   0.561591   |   0.596576   |  -4.846302   |-1.006628e+01 |-7.423447e-01 |
|  620  |15.5000 |  1   |   0.559380   |   0.594044   |  -4.846417   |-1.006400e+01 |-7.425361e-01 |
|  621  |15.5250 |  1   |   0.550661   |   0.639299   |  -4.846521   |-1.006172e+01 |-7.427276e-01 |
|  622  |15.5500 |  1   |   0.564344   |   0.599497   |  -4.846664   |-1.005944e+01 |-7.429191e-01 |
|  623  |15.5750 |  1   |   0.564470   |   0.598390   |  -4.846375   |-1.005716e+01 |-7.431107e-01 |
|  624  |15.6000 |  1   |   0.582095   |   0.621527   |  -4.846646   |-1.005489e+01 |-7.433022e-01 |
|  625  |15.6250 |  1   |   0.563067   |   0.594485   |  -4.846450   |-1.005262e+01 |-7.434937e-01 |
|  626  |15.6500 |  1   |   0.556217   |   0.597713   |  -4.847057   |-1.005035e+01 |-7.436852e-01 |
|  627  |15.6750 |  1   |   0.556730   |   0.599759   |  -4.847059   |-1.004809e+01 |-7.438766e-01 |
|  628  |15.7000 |  1   |   0.559253   |   0.596950   |  -4.847442   |-1.004582e+01 |-7.440680e-01 |
|  629  |15.7250 |  1   |   0.566564   |   0.599786   |  -4.847356   |-1.004357e+01 |-7.442594e-01 |
|  630  |15.7500 |  1   |   0.541901   |   0.610011   |  -4.847580   |-1.004131e+01 |-7.444507e-01 |
|  631  |15.7750 |  1   |   0.552107   |   0.603683   |  -4.847782   |-1.003906e+01 |-7.446419e-01 |
|  632  |15.8000 |  1   |   0.559163   |   0.600536   |  -4.848163   |-1.003681e+01 |-7.448331e-01 |
|  633  |15.8250 |  1   |   0.565520   |   0.607117   |  -4.848649   |-1.003456e+01 |-7.450241e-01 |
|  634  |15.8500 |  1   |   0.563058   |   0.602004   |  -4.848834   |-1.003232e+01 |-7.452150e-01 |
|  635  |15.8750 |  1   |   0.565077   |   0.604592   |  -4.849242   |-1.003009e+01 |-7.454058e-01 |
|  636  |15.9000 |  1   |   0.561084   |   0.613315   |  -4.849257   |-1.002785e+01 |-7.455965e-01 |
|  637  |15.9250 |  1   |   0.562131   |   0.608055   |  -4.849893   |-1.002562e+01 |-7.457870e-01 |
|  638  |15.9500 |  1   |   0.567037   |   0.607231   |  -4.850162   |-1.002340e+01 |-7.459774e-01 |
|  639  |15.9750 |  1   |   0.559362   |   0.605681   |  -4.850853   |-1.002118e+01 |-7.461676e-01 |
|  640  |16.0000 |  1   |   0.559990   |   0.606771   |  -4.851290   |-1.001896e+01 |-7.463577e-01 |
|  641  |16.0250 |  1   |   0.563540   |   0.604675   |  -4.851567   |-1.001675e+01 |-7.465476e-01 |
|  642  |16.0500 |  1   |   0.569776   |   0.614118   |  -4.852131   |-1.001455e+01 |-7.467372e-01 |
|  643  |16.0750 |  1   |   0.526022   |   0.638151   |  -4.852404   |-1.001234e+01 |-7.469267e-01 |
|  644  |16.1000 |  1   |   0.561008   |   0.613815   |  -4.853264   |-1.001015e+01 |-7.471159e-01 |
|  645  |16.1250 |  1   |   0.558149   |   0.610124   |  -4.853699   |-1.000796e+01 |-7.473050e-01 |
|  646  |16.1500 |  1   |   0.558817   |   0.602424   |  -4.854509   |-1.000577e+01 |-7.474938e-01 |
|  647  |16.1750 |  1   |   0.551800   |   0.649783   |  -4.854874   |-1.000359e+01 |-7.476823e-01 |
|  648  |16.2000 |  1   |   0.563560   |   0.614133   |  -4.830412   |-1.000141e+01 |-7.478702e-01 |
|  649  |16.2250 |  1   |   0.527692   |   0.637038   |  -4.860230   |-9.999242e+00 |-7.480574e-01 |
|  650  |16.2500 |  1   |   0.520572   |   0.630335   |  -4.856936   |-9.997077e+00 |-7.482449e-01 |
|  651  |16.2750 |  1   |   0.538065   |   0.623007   |  -4.855762   |-9.994918e+00 |-7.484321e-01 |
|  652  |16.3000 |  1   |   0.563850   |   0.606399   |  -4.859225   |-9.992765e+00 |-7.486189e-01 |
|  653  |16.3250 |  1   |   0.569525   |   0.613083   |  -4.857579   |-9.990618e+00 |-7.488056e-01 |
|  654  |16.3500 |  1   |   0.572920   |   0.616659   |  -4.856411   |-9.988477e+00 |-7.489920e-01 |
|  655  |16.3750 |  1   |   0.565054   |   0.622951   |  -4.859712   |-9.986343e+00 |-7.491781e-01 |
|  656  |16.4000 |  1   |   0.538484   |   0.637184   |  -4.858656   |-9.984214e+00 |-7.493640e-01 |
|  657  |16.4250 |  1   |   0.572575   |   0.628920   |  -4.863059   |-9.982092e+00 |-7.495496e-01 |
|  658  |16.4500 |  1   |   0.553086   |   0.615147   |  -4.865486   |-9.979977e+00 |-7.497349e-01 |
|  659  |16.4750 |  1   |   0.554584   |   0.608671   |  -4.862494   |-9.977868e+00 |-7.499197e-01 |
|  660  |16.5000 |  1   |   0.566182   |   0.632013   |  -4.865031   |-9.975766e+00 |-7.501042e-01 |
|  661  |16.5250 |  1   |   0.559237   |   0.607847   |  -4.864222   |-9.973671e+00 |-7.502883e-01 |
|  662  |16.5500 |  1   |   0.547495   |   0.609668   |  -4.865897   |-9.971582e+00 |-7.504721e-01 |
|  663  |16.5750 |  1   |   0.554912   |   0.604920   |  -4.870242   |-9.969500e+00 |-7.506554e-01 |
|  664  |16.6000 |  1   |   0.563164   |   0.608630   |  -4.868999   |-9.967426e+00 |-7.508382e-01 |
|  665  |16.6250 |  1   |   0.557828   |   0.607771   |  -4.870617   |-9.965359e+00 |-7.510206e-01 |
|  666  |16.6500 |  1   |   0.564567   |   0.611154   |  -4.869702   |-9.963299e+00 |-7.512027e-01 |
|  667  |16.6750 |  1   |   0.570958   |   0.634616   |  -4.868988   |-9.961247e+00 |-7.513841e-01 |
|  668  |16.7000 |  1   |   0.555363   |   0.608414   |  -4.873698   |-9.959202e+00 |-7.515652e-01 |
|  669  |16.7250 |  1   |   0.550943   |   0.608364   |  -4.874542   |-9.957164e+00 |-7.517458e-01 |
|  670  |16.7500 |  1   |   0.555703   |   0.608518   |  -4.875535   |-9.955134e+00 |-7.519257e-01 |
|  671  |16.7750 |  1   |   0.545052   |   0.613267   |  -4.877426   |-9.953112e+00 |-7.521053e-01 |
|  672  |16.8000 |  1   |   0.553070   |   0.605358   |  -4.874922   |-9.951098e+00 |-7.522844e-01 |
|  673  |16.8250 |  1   |   0.561627   |   0.612107   |  -4.876898   |-9.949092e+00 |-7.524629e-01 |
|  674  |16.8500 |  1   |   0.555659   |   0.607956   |  -4.879987   |-9.947094e+00 |-7.526409e-01 |
|  675  |16.8750 |  1   |   0.553403   |   0.616417   |  -4.880702   |-9.945105e+00 |-7.528184e-01 |
|  676  |16.9000 |  1   |   0.553432   |   0.612899   |  -4.884087   |-9.943123e+00 |-7.529953e-01 |
|  677  |16.9250 |  1   |   0.559188   |   0.618660   |  -4.883577   |-9.941150e+00 |-7.531718e-01 |
|  678  |16.9500 |  1   |   0.555787   |   0.615341   |  -4.883057   |-9.939185e+00 |-7.533476e-01 |
|  679  |16.9750 |  1   |   0.558311   |   0.612354   |  -4.885587   |-9.937229e+00 |-7.535229e-01 |
|  680  |17.0000 |  1   |   0.561945   |   0.618028   |  -4.886378   |-9.935282e+00 |-7.536977e-01 |
|  681  |17.0250 |  1   |   0.539489   |   0.628249   |  -4.890029   |-9.933343e+00 |-7.538718e-01 |
|  682  |17.0500 |  1   |   0.514794   |   0.643622   |  -4.892106   |-9.931413e+00 |-7.540454e-01 |
|  683  |17.0750 |  1   |   0.539393   |   0.626948   |  -4.890958   |-9.929493e+00 |-7.542183e-01 |
|  684  |17.1000 |  1   |   0.560875   |   0.618428   |  -4.893438   |-9.927581e+00 |-7.543907e-01 |
|  685  |17.1250 |  1   |   0.554366   |   0.611880   |  -4.893277   |-9.925678e+00 |-7.545624e-01 |
|  686  |17.1500 |  1   |   0.553763   |   0.618234   |  -4.894869   |-9.923784e+00 |-7.547335e-01 |
|  687  |17.1750 |  1   |   0.553390   |   0.611751   |  -4.899919   |-9.921900e+00 |-7.549040e-01 |
|  688  |17.2000 |  1   |   0.556929   |   0.626983   |  -4.899359   |-9.920025e+00 |-7.550737e-01 |
|  689  |17.2250 |  1   |   0.551784   |   0.617594   |  -4.901290   |-9.918159e+00 |-7.552428e-01 |
|  690  |17.2500 |  1   |   0.534141   |   0.623701   |  -4.902376   |-9.916303e+00 |-7.554113e-01 |
|  691  |17.2750 |  1   |   0.520007   |   0.638176   |  -4.901707   |-9.914456e+00 |-7.555791e-01 |
|  692  |17.3000 |  1   |   0.514522   |   0.638778   |  -4.905930   |-9.912619e+00 |-7.557462e-01 |
|  693  |17.3250 |  1   |   0.510692   |   0.643761   |  -4.907376   |-9.910792e+00 |-7.559126e-01 |
|  694  |17.3500 |  1   |   0.517713   |   0.643936   |  -4.909717   |-9.908974e+00 |-7.560782e-01 |
|  695  |17.3750 |  1   |   0.541182   |   0.655376   |  -4.911696   |-9.907166e+00 |-7.562432e-01 |
|  696  |17.4000 |  1   |   0.551757   |   0.611377   |  -4.910876   |-9.905369e+00 |-7.564075e-01 |
|  697  |17.4250 |  1   |   0.555262   |   0.614891   |  -4.913403   |-9.903581e+00 |-7.565710e-01 |
|  698  |17.4500 |  1   |   0.564454   |   0.624082   |  -4.915500   |-9.901804e+00 |-7.567338e-01 |
|  699  |17.4750 |  1   |   0.553199   |   0.610732   |  -4.917215   |-9.900037e+00 |-7.568959e-01 |
|  700  |17.5000 |  1   |   0.548138   |   0.608044   |  -4.921356   |-9.898280e+00 |-7.570572e-01 |
|  701  |17.5250 |  1   |   0.549388   |   0.607808   |  -4.921642   |-9.896533e+00 |-7.572177e-01 |
|  702  |17.5500 |  1   |   0.553081   |   0.618402   |  -4.922276   |-9.894797e+00 |-7.573775e-01 |
|  703  |17.5750 |  1   |   0.551544   |   0.613274   |  -4.925124   |-9.893071e+00 |-7.575366e-01 |
|  704  |17.6000 |  1   |   0.537771   |   0.622332   |  -4.925960   |-9.891356e+00 |-7.576948e-01 |
|  705  |17.6250 |  1   |   0.547941   |   0.612396   |  -4.929428   |-9.889651e+00 |-7.578523e-01 |
|  706  |17.6500 |  1   |   0.558546   |   0.618044   |  -4.932578   |-9.887957e+00 |-7.580090e-01 |
|  707  |17.6750 |  1   |   0.550670   |   0.613128   |  -4.933198   |-9.886274e+00 |-7.581648e-01 |
|  708  |17.7000 |  1   |   0.549359   |   0.610619   |  -4.935304   |-9.884602e+00 |-7.583199e-01 |
|  709  |17.7250 |  1   |   0.548109   |   0.608598   |  -4.936363   |-9.882941e+00 |-7.584741e-01 |
|  710  |17.7500 |  1   |   0.550066   |   0.621723   |  -4.938473   |-9.881290e+00 |-7.586276e-01 |
|  711  |17.7750 |  1   |   0.547227   |   0.613100   |  -4.942387   |-9.879651e+00 |-7.587801e-01 |
|  712  |17.8000 |  1   |   0.549032   |   0.607084   |  -4.943841   |-9.878023e+00 |-7.589318e-01 |
|  713  |17.8250 |  1   |   0.553408   |   0.626393   |  -4.946716   |-9.876405e+00 |-7.590827e-01 |
|  714  |17.8500 |  1   |   0.552940   |   0.612425   |  -4.948343   |-9.874799e+00 |-7.592327e-01 |
|  715  |17.8750 |  1   |   0.547835   |   0.608350   |  -4.948430   |-9.873205e+00 |-7.593819e-01 |
|  716  |17.9000 |  1   |   0.540325   |   0.615331   |  -4.952754   |-9.871621e+00 |-7.595302e-01 |
|  717  |17.9250 |  1   |   0.541983   |   0.615043   |  -4.954732   |-9.870049e+00 |-7.596776e-01 |
|  718  |17.9500 |  1   |   0.545322   |   0.613781   |  -4.957229   |-9.868489e+00 |-7.598241e-01 |
|  719  |17.9750 |  1   |   0.550283   |   0.619134   |  -4.960988   |-9.866940e+00 |-7.599697e-01 |
|  720  |18.0000 |  1   |   0.545893   |   0.614056   |  -4.961059   |-9.865402e+00 |-7.601144e-01 |
|  721  |18.0250 |  1   |   0.545672   |   0.612643   |  -4.963571   |-9.863876e+00 |-7.602583e-01 |
|  722  |18.0500 |  1   |   0.547688   |   0.613754   |  -4.966223   |-9.862362e+00 |-7.604012e-01 |
|  723  |18.0750 |  1   |   0.552767   |   0.618856   |  -4.968695   |-9.860860e+00 |-7.605432e-01 |
|  724  |18.1000 |  1   |   0.536194   |   0.619256   |  -4.972596   |-9.859369e+00 |-7.606842e-01 |
|  725  |18.1250 |  1   |   0.547934   |   0.614264   |  -4.974514   |-9.857890e+00 |-7.608244e-01 |
|  726  |18.1500 |  1   |   0.546250   |   0.609647   |  -4.976536   |-9.856423e+00 |-7.609636e-01 |
|  727  |18.1750 |  1   |   0.543682   |   0.612869   |  -4.979007   |-9.854968e+00 |-7.611018e-01 |
|  728  |18.2000 |  1   |   0.540415   |   0.613370   |  -4.980559   |-9.853525e+00 |-7.612392e-01 |
|  729  |18.2250 |  1   |   0.545937   |   0.614889   |  -4.984680   |-9.852095e+00 |-7.613755e-01 |
|  730  |18.2500 |  1   |   0.545980   |   0.617822   |  -4.988031   |-9.850676e+00 |-7.615109e-01 |
|  731  |18.2750 |  1   |   0.542477   |   0.615399   |  -4.989711   |-9.849269e+00 |-7.616453e-01 |
|  732  |18.3000 |  1   |   0.547334   |   0.637717   |  -4.993188   |-9.847874e+00 |-7.617788e-01 |
|  733  |18.3250 |  1   |   0.553981   |   0.623099   |  -4.994720   |-9.846492e+00 |-7.619113e-01 |
|  734  |18.3500 |  1   |   0.548177   |   0.615461   |  -4.996885   |-9.845122e+00 |-7.620428e-01 |
|  735  |18.3750 |  1   |   0.541814   |   0.619284   |  -5.001297   |-9.843764e+00 |-7.621732e-01 |
|  736  |18.4000 |  1   |   0.545819   |   0.612507   |  -5.004011   |-9.842419e+00 |-7.623028e-01 |
|  737  |18.4250 |  1   |   0.544200   |   0.612841   |  -5.007008   |-9.841085e+00 |-7.624312e-01 |
|  738  |18.4500 |  1   |   0.542160   |   0.613612   |  -5.009954   |-9.839765e+00 |-7.625587e-01 |
|  739  |18.4750 |  1   |   0.579838   |   0.652729   |  -5.011577   |-9.838457e+00 |-7.626852e-01 |
|  740  |18.5000 |  1   |   0.502843   |   0.647341   |  -5.014921   |-9.837161e+00 |-7.628107e-01 |
|  741  |18.5250 |  1   |   0.551553   |   0.615329   |  -5.018108   |-9.835878e+00 |-7.629351e-01 |
|  742  |18.5500 |  1   |   0.527083   |   0.626063   |  -5.021511   |-9.834607e+00 |-7.630585e-01 |
|  743  |18.5750 |  1   |   0.539414   |   0.623069   |  -5.025618   |-9.833349e+00 |-7.631809e-01 |
|  744  |18.6000 |  1   |   0.525490   |   0.626166   |  -5.027066   |-9.832104e+00 |-7.633022e-01 |
|  745  |18.6250 |  1   |   0.517001   |   0.654233   |  -5.030386   |-9.830872e+00 |-7.634225e-01 |
|  746  |18.6500 |  1   |   0.534137   |   0.626574   |  -5.033620   |-9.829652e+00 |-7.635418e-01 |
|  747  |18.6750 |  1   |   0.542476   |   0.619676   |  -5.035894   |-9.828445e+00 |-7.636599e-01 |
|  748  |18.7000 |  1   |   0.553023   |   0.625866   |  -5.041104   |-9.827250e+00 |-7.637771e-01 |
|  749  |18.7250 |  1   |   0.540949   |   0.619120   |  -5.044064   |-9.826069e+00 |-7.638931e-01 |
|  750  |18.7500 |  1   |   0.537734   |   0.621421   |  -5.046464   |-9.824900e+00 |-7.640081e-01 |
|  751  |18.7750 |  1   |   0.538831   |   0.623689   |  -5.050195   |-9.823745e+00 |-7.641221e-01 |
|  752  |18.8000 |  1   |   0.541038   |   0.621485   |  -5.052620   |-9.822602e+00 |-7.642350e-01 |
|  753  |18.8250 |  1   |   0.535511   |   0.620073   |  -5.056418   |-9.821472e+00 |-7.643467e-01 |
|  754  |18.8500 |  1   |   0.530924   |   0.626769   |  -5.060668   |-9.820355e+00 |-7.644574e-01 |
|  755  |18.8750 |  1   |   0.537222   |   0.615511   |  -5.064038   |-9.819251e+00 |-7.645670e-01 |
|  756  |18.9000 |  1   |   0.534256   |   0.619716   |  -5.067645   |-9.818160e+00 |-7.646756e-01 |
|  757  |18.9250 |  1   |   0.535977   |   0.618753   |  -5.070290   |-9.817082e+00 |-7.647830e-01 |
|  758  |18.9500 |  1   |   0.540988   |   0.623975   |  -5.073509   |-9.816018e+00 |-7.648893e-01 |
|  759  |18.9750 |  1   |   0.549132   |   0.621750   |  -5.078004   |-9.814966e+00 |-7.649945e-01 |
|  760  |19.0000 |  1   |   0.539145   |   0.617640   |  -5.081158   |-9.813927e+00 |-7.650987e-01 |
|  761  |19.0250 |  1   |   0.540759   |   0.626675   |  -5.085734   |-9.812902e+00 |-7.652017e-01 |
|  762  |19.0500 |  1   |   0.538078   |   0.622588   |  -5.089589   |-9.811890e+00 |-7.653036e-01 |
|  763  |19.0750 |  1   |   0.522208   |   0.636657   |  -5.091690   |-9.810890e+00 |-7.654044e-01 |
|  764  |19.1000 |  1   |   0.533418   |   0.625254   |  -5.096080   |-9.809904e+00 |-7.655040e-01 |
|  765  |19.1250 |  1   |   0.539772   |   0.619554   |  -5.100005   |-9.808932e+00 |-7.656026e-01 |
|  766  |19.1500 |  1   |   0.539060   |   0.621184   |  -5.103610   |-9.807972e+00 |-7.657000e-01 |
|  767  |19.1750 |  1   |   0.535861   |   0.620145   |  -5.108621   |-9.807026e+00 |-7.657963e-01 |
|  768  |19.2000 |  1   |   0.536273   |   0.636314   |  -5.112126   |-9.806093e+00 |-7.658914e-01 |
|  769  |19.2250 |  1   |   0.527759   |   0.633821   |  -5.115285   |-9.805173e+00 |-7.659854e-01 |
|  770  |19.2500 |  1   |   0.544239   |   0.628317   |  -5.119466   |-9.804266e+00 |-7.660783e-01 |
|  771  |19.2750 |  1   |   0.537259   |   0.622405   |  -5.123024   |-9.803373e+00 |-7.661701e-01 |
|  772  |19.3000 |  1   |   0.534405   |   0.621513   |  -5.128168   |-9.802493e+00 |-7.662607e-01 |
|  773  |19.3250 |  1   |   0.536213   |   0.619574   |  -5.132116   |-9.801626e+00 |-7.663501e-01 |
|  774  |19.3500 |  1   |   0.547262   |   0.633010   |  -5.136146   |-9.800773e+00 |-7.664384e-01 |
|  775  |19.3750 |  1   |   0.532413   |   0.624157   |  -5.140640   |-9.799933e+00 |-7.665256e-01 |
|  776  |19.4000 |  1   |   0.528069   |   0.631161   |  -5.143361   |-9.799106e+00 |-7.666116e-01 |
|  777  |19.4250 |  1   |   0.508258   |   0.655418   |  -5.148293   |-9.798293e+00 |-7.666965e-01 |
|  778  |19.4500 |  1   |   0.531947   |   0.619965   |  -5.153388   |-9.797493e+00 |-7.667802e-01 |
|  779  |19.4750 |  1   |   0.538257   |   0.624263   |  -5.157054   |-9.796706e+00 |-7.668627e-01 |
|  780  |19.5000 |  1   |   0.513675   |   0.639047   |  -5.162093   |-9.795933e+00 |-7.669441e-01 |
|  781  |19.5250 |  1   |   0.524592   |   0.629631   |  -5.166237   |-9.795173e+00 |-7.670243e-01 |
|  782  |19.5500 |  1   |   0.501278   |   0.646392   |  -5.169515   |-9.794426e+00 |-7.671034e-01 |
|  783  |19.5750 |  1   |   0.508310   |   0.653496   |  -5.174678   |-9.793693e+00 |-7.671812e-01 |
|  784  |19.6000 |  1   |   0.501368   |   0.644803   |  -5.179349   |-9.792973e+00 |-7.672580e-01 |
|  785  |19.6250 |  1   |   0.505631   |   0.653741   |  -5.184141   |-9.792267e+00 |-7.673335e-01 |
|  786  |19.6500 |  1   |   0.501214   |   0.647573   |  -5.188924   |-9.791574e+00 |-7.674079e-01 |
|  787  |19.6750 |  1   |   0.495648   |   0.655176   |  -5.192845   |-9.790894e+00 |-7.674811e-01 |
|  788  |19.7000 |  1   |   0.499932   |   0.647758   |  -5.197708   |-9.790228e+00 |-7.675532e-01 |
|  789  |19.7250 |  1   |   0.518200   |   0.630863   |  -5.201669   |-9.789575e+00 |-7.676241e-01 |
|  790  |19.7500 |  1   |   0.536332   |   0.619421   |  -5.206930   |-9.788935e+00 |-7.676937e-01 |
|  791  |19.7750 |  1   |   0.531387   |   0.621932   |  -5.212611   |-9.788309e+00 |-7.677623e-01 |
|  792  |19.8000 |  1   |   0.527384   |   0.622412   |  -5.216559   |-9.787696e+00 |-7.678296e-01 |
|  793  |19.8250 |  1   |   0.529678   |   0.620205   |  -5.221430   |-9.787096e+00 |-7.678958e-01 |
|  794  |19.8500 |  1   |   0.508913   |   0.648031   |  -5.226400   |-9.786510e+00 |-7.679608e-01 |
|  795  |19.8750 |  1   |   0.504351   |   0.648719   |  -5.230347   |-9.785937e+00 |-7.680246e-01 |
|  796  |19.9000 |  1   |   0.500118   |   0.648664   |  -5.235880   |-9.785377e+00 |-7.680872e-01 |
|  797  |19.9250 |  1   |   0.528246   |   0.630585   |  -5.241694   |-9.784831e+00 |-7.681487e-01 |
|  798  |19.9500 |  1   |   0.542162   |   0.629759   |  -5.245912   |-9.784298e+00 |-7.682090e-01 |
|  799  |19.9750 |  1   |   0.529981   |   0.619840   |  -5.251163   |-9.783778e+00 |-7.682681e-01 |
|  800  |20.0000 |  1   |   0.529003   |   0.618461   |  -5.255740   |-9.783271e+00 |-7.683260e-01 |
|  801  |20.0250 |  1   |   0.531965   |   0.615729   |  -5.260634   |-9.782778e+00 |-7.683827e-01 |
|  802  |20.0500 |  1   |   0.535846   |   0.625167   |  -5.265935   |-9.782298e+00 |-7.684383e-01 |
|  803  |20.0750 |  1   |   0.530973   |   0.621185   |  -5.271230   |-9.781831e+00 |-7.684927e-01 |
|  804  |20.1000 |  1   |   0.534419   |   0.622091   |  -5.277057   |-9.781378e+00 |-7.685459e-01 |
|  805  |20.1250 |  1   |   0.534982   |   0.622699   |  -5.281286   |-9.780938e+00 |-7.685979e-01 |
|  806  |20.1500 |  1   |   0.530764   |   0.620805   |  -5.286196   |-9.780510e+00 |-7.686487e-01 |
|  807  |20.1750 |  1   |   0.511985   |   0.636388   |  -5.291943   |-9.780096e+00 |-7.686984e-01 |
|  808  |20.2000 |  1   |   0.496395   |   0.650078   |  -5.296389   |-9.779696e+00 |-7.687469e-01 |
|  809  |20.2250 |  1   |   0.500372   |   0.658939   |  -5.302208   |-9.779308e+00 |-7.687942e-01 |
|  810  |20.2500 |  1   |   0.499443   |   0.650036   |  -5.308191   |-9.778933e+00 |-7.688403e-01 |
|  811  |20.2750 |  1   |   0.507094   |   0.660675   |  -5.312377   |-9.778572e+00 |-7.688852e-01 |
|  812  |20.3000 |  1   |   0.501244   |   0.655777   |  -5.317533   |-9.778224e+00 |-7.689290e-01 |
|  813  |20.3250 |  1   |   0.497886   |   0.659485   |  -5.322998   |-9.777888e+00 |-7.689716e-01 |
|  814  |20.3500 |  1   |   0.531331   |   0.625960   |  -5.327874   |-9.777566e+00 |-7.690130e-01 |
|  815  |20.3750 |  1   |   0.520818   |   0.633773   |  -5.333552   |-9.777257e+00 |-7.690533e-01 |
|  816  |20.4000 |  1   |   0.528115   |   0.629487   |  -5.338954   |-9.776961e+00 |-7.690924e-01 |
|  817  |20.4250 |  1   |   0.533945   |   0.639602   |  -5.344271   |-9.776677e+00 |-7.691303e-01 |
|  818  |20.4500 |  1   |   0.518692   |   0.616616   |  -5.348659   |-9.776407e+00 |-7.691670e-01 |
|  819  |20.4750 |  1   |   0.522226   |   0.616927   |  -5.353802   |-9.776149e+00 |-7.692026e-01 |
|  820  |20.5000 |  1   |   0.488609   |   0.653811   |  -5.359571   |-9.775905e+00 |-7.692370e-01 |
|  821  |20.5250 |  1   |   0.484709   |   0.644598   |  -5.364204   |-9.775673e+00 |-7.692702e-01 |
|  822  |20.5500 |  1   |   0.486915   |   0.644077   |  -5.369689   |-9.775454e+00 |-7.693023e-01 |
|  823  |20.5750 |  1   |   0.486652   |   0.644571   |  -5.375147   |-9.775248e+00 |-7.693332e-01 |
|  824  |20.6000 |  1   |   0.504305   |   0.636577   |  -5.379300   |-9.775055e+00 |-7.693629e-01 |
|  825  |20.6250 |  1   |   0.498041   |   0.634625   |  -5.383927   |-9.774875e+00 |-7.693915e-01 |
|  826  |20.6500 |  1   |   0.491458   |   0.658197   |  -5.389699   |-9.774707e+00 |-7.694190e-01 |
|  827  |20.6750 |  1   |   0.496825   |   0.658731   |  -5.394111   |-9.774552e+00 |-7.694452e-01 |
|  828  |20.7000 |  1   |   0.502448   |   0.654295   |  -5.398969   |-9.774409e+00 |-7.694703e-01 |
|  829  |20.7250 |  1   |   0.526343   |   0.629665   |  -5.404191   |-9.774279e+00 |-7.694943e-01 |
|  830  |20.7500 |  1   |   0.527985   |   0.637049   |  -5.408232   |-9.774162e+00 |-7.695171e-01 |
|  831  |20.7750 |  1   |   0.526428   |   0.631766   |  -5.412435   |-9.774057e+00 |-7.695388e-01 |
|  832  |20.8000 |  1   |   0.549858   |   0.656701   |  -5.417086   |-9.773965e+00 |-7.695593e-01 |
|  833  |20.8250 |  1   |   0.525367   |   0.640356   |  -5.421889   |-9.773886e+00 |-7.695787e-01 |
|  834  |20.8500 |  1   |   0.526791   |   0.632546   |  -5.425807   |-9.773818e+00 |-7.695970e-01 |
|  835  |20.8750 |  1   |   0.527689   |   0.634832   |  -5.429904   |-9.773763e+00 |-7.696141e-01 |
|  836  |20.9000 |  1   |   0.527279   |   0.632626   |  -5.434402   |-9.773721e+00 |-7.696301e-01 |
|  837  |20.9250 |  1   |   0.522657   |   0.645628   |  -5.437396   |-9.773691e+00 |-7.696449e-01 |
|  838  |20.9500 |  1   |   0.525489   |   0.635935   |  -5.441073   |-9.773673e+00 |-7.696586e-01 |
|  839  |20.9750 |  1   |   0.532311   |   0.650099   |  -5.445642   |-9.773668e+00 |-7.696712e-01 |
|  840  |21.0000 |  1   |   0.529552   |   0.651858   |  -5.448456   |-9.773674e+00 |-7.696827e-01 |
|  841  |21.0250 |  1   |   0.503458   |   0.671841   |  -5.451750   |-9.773693e+00 |-7.696930e-01 |
|  842  |21.0500 |  1   |   0.501758   |   0.675092   |  -5.455281   |-9.773724e+00 |-7.697022e-01 |
|  843  |21.0750 |  1   |   0.514197   |   0.677148   |  -5.457769   |-9.773768e+00 |-7.697103e-01 |
|  844  |21.1000 |  1   |   0.501449   |   0.669868   |  -5.460336   |-9.773823e+00 |-7.697173e-01 |
|  845  |21.1250 |  1   |   0.497707   |   0.677975   |  -5.463272   |-9.773890e+00 |-7.697232e-01 |
|  846  |21.1500 |  1   |   0.500228   |   0.670709   |  -5.466064   |-9.773969e+00 |-7.697280e-01 |
|  847  |21.1750 |  1   |   0.504896   |   0.676630   |  -5.467633   |-9.774060e+00 |-7.697316e-01 |
|  848  |21.2000 |  1   |   0.500956   |   0.671256   |  -5.469894   |-9.774164e+00 |-7.697342e-01 |
|  849  |21.2250 |  1   |   0.496505   |   0.674550   |  -5.472033   |-9.774278e+00 |-7.697357e-01 |
|  850  |21.2500 |  1   |   0.529665   |   0.644782   |  -5.472924   |-9.774405e+00 |-7.697361e-01 |
|  851  |21.2750 |  1   |   0.528408   |   0.645465   |  -5.474619   |-9.774544e+00 |-7.697354e-01 |
|  852  |21.3000 |  1   |   0.519741   |   0.635947   |  -5.476252   |-9.774694e+00 |-7.697336e-01 |
|  853  |21.3250 |  1   |   0.504346   |   0.651090   |  -5.476890   |-9.774856e+00 |-7.697307e-01 |
|  854  |21.3500 |  1   |   0.532271   |   0.638573   |  -5.477458   |-9.775029e+00 |-7.697268e-01 |
|  855  |21.3750 |  1   |   0.523634   |   0.638200   |  -5.478494   |-9.775214e+00 |-7.697218e-01 |
|  856  |21.4000 |  1   |   0.519422   |   0.638265   |  -5.478639   |-9.775411e+00 |-7.697157e-01 |
|  857  |21.4250 |  1   |   0.518510   |   0.639035   |  -5.478368   |-9.775619e+00 |-7.697085e-01 |
|  858  |21.4500 |  1   |   0.510864   |   0.672835   |  -5.478909   |-9.775839e+00 |-7.697003e-01 |
|  859  |21.4750 |  1   |   0.524935   |   0.637128   |  -5.478554   |-9.776069e+00 |-7.696910e-01 |
|  860  |21.5000 |  1   |   0.507660   |   0.648373   |  -5.477690   |-9.776312e+00 |-7.696807e-01 |
|  861  |21.5250 |  1   |   0.499457   |   0.665963   |  -5.477335   |-9.776565e+00 |-7.696693e-01 |
|  862  |21.5500 |  1   |   0.499270   |   0.658582   |  -5.476603   |-9.776830e+00 |-7.696569e-01 |
|  863  |21.5750 |  1   |   0.519785   |   0.638694   |  -5.475176   |-9.777106e+00 |-7.696434e-01 |
|  864  |21.6000 |  1   |   0.534959   |   0.658340   |  -5.473925   |-9.777393e+00 |-7.696289e-01 |
|  865  |21.6250 |  1   |   0.516683   |   0.652932   |  -5.472998   |-9.777691e+00 |-7.696133e-01 |
|  866  |21.6500 |  1   |   0.524993   |   0.643355   |  -5.470927   |-9.778000e+00 |-7.695968e-01 |
|  867  |21.6750 |  1   |   0.521338   |   0.641387   |  -5.469193   |-9.778320e+00 |-7.695792e-01 |
|  868  |21.7000 |  1   |   0.511949   |   0.650276   |  -5.467707   |-9.778651e+00 |-7.695606e-01 |
|  869  |21.7250 |  1   |   0.516787   |   0.641368   |  -5.465221   |-9.778993e+00 |-7.695409e-01 |
|  870  |21.7500 |  1   |   0.526370   |   0.644767   |  -5.462966   |-9.779345e+00 |-7.695203e-01 |
|  871  |21.7750 |  1   |   0.527064   |   0.650980   |  -5.460746   |-9.779709e+00 |-7.694986e-01 |
|  872  |21.8000 |  1   |   0.516300   |   0.660074   |  -5.458348   |-9.780083e+00 |-7.694760e-01 |
|  873  |21.8250 |  1   |   0.516653   |   0.640115   |  -5.455473   |-9.780468e+00 |-7.694523e-01 |
|  874  |21.8500 |  1   |   0.519911   |   0.637954   |  -5.452906   |-9.780863e+00 |-7.694277e-01 |
|  875  |21.8750 |  1   |   0.515326   |   0.653226   |  -5.450254   |-9.781269e+00 |-7.694021e-01 |
|  876  |21.9000 |  1   |   0.519997   |   0.639056   |  -5.446816   |-9.781685e+00 |-7.693755e-01 |
|  877  |21.9250 |  1   |   0.520825   |   0.640152   |  -5.443891   |-9.782112e+00 |-7.693479e-01 |
|  878  |21.9500 |  1   |   0.517811   |   0.643432   |  -5.441052   |-9.782549e+00 |-7.693193e-01 |
|  879  |21.9750 |  1   |   0.514777   |   0.645231   |  -5.437508   |-9.782996e+00 |-7.692898e-01 |
|  880  |22.0000 |  1   |   0.509819   |   0.664910   |  -5.434299   |-9.783453e+00 |-7.692593e-01 |
|  881  |22.0250 |  1   |   0.533964   |   0.644499   |  -5.431166   |-9.783921e+00 |-7.692279e-01 |
|  882  |22.0500 |  1   |   0.515048   |   0.643263   |  -5.427340   |-9.784399e+00 |-7.691955e-01 |
|  883  |22.0750 |  1   |   0.528883   |   0.638636   |  -5.423886   |-9.784887e+00 |-7.691622e-01 |
|  884  |22.1000 |  1   |   0.520719   |   0.634711   |  -5.420468   |-9.785385e+00 |-7.691279e-01 |
|  885  |22.1250 |  1   |   0.520284   |   0.649912   |  -5.416873   |-9.785892e+00 |-7.690927e-01 |
|  886  |22.1500 |  1   |   0.528762   |   0.647822   |  -5.413226   |-9.786410e+00 |-7.690565e-01 |
|  887  |22.1750 |  1   |   0.509786   |   0.659574   |  -5.409500   |-9.786937e+00 |-7.690194e-01 |
|  888  |22.2000 |  1   |   0.520104   |   0.640645   |  -5.405926   |-9.787475e+00 |-7.689815e-01 |
|  889  |22.2250 |  1   |   0.528201   |   0.646174   |  -5.401876   |-9.788022e+00 |-7.689426e-01 |
|  890  |22.2500 |  1   |   0.517401   |   0.639639   |  -5.398218   |-9.788578e+00 |-7.689027e-01 |
|  891  |22.2750 |  1   |   0.520639   |   0.632424   |  -5.394694   |-9.789144e+00 |-7.688620e-01 |
|  892  |22.3000 |  1   |   0.522067   |   0.636225   |  -5.390716   |-9.789720e+00 |-7.688204e-01 |
|  893  |22.3250 |  1   |   0.517790   |   0.632747   |  -5.386888   |-9.790305e+00 |-7.687779e-01 |
|  894  |22.3500 |  1   |   0.507874   |   0.644954   |  -5.383266   |-9.790899e+00 |-7.687345e-01 |
|  895  |22.3750 |  1   |   0.464989   |   0.691658   |  -5.379134   |-9.791502e+00 |-7.686902e-01 |
|  896  |22.4000 |  1   |   0.480783   |   0.677711   |  -5.375381   |-9.792115e+00 |-7.686451e-01 |
|  897  |22.4250 |  1   |   0.490632   |   0.674024   |  -5.371811   |-9.792737e+00 |-7.685991e-01 |
|  898  |22.4500 |  1   |   0.481621   |   0.671897   |  -5.367827   |-9.793368e+00 |-7.685522e-01 |
|  899  |22.4750 |  1   |   0.474796   |   0.677255   |  -5.364149   |-9.794009e+00 |-7.685045e-01 |
|  900  |22.5000 |  1   |   0.491788   |   0.670256   |  -5.360229   |-9.794658e+00 |-7.684559e-01 |
|  901  |22.5250 |  1   |   0.484136   |   0.663273   |  -5.356469   |-9.795316e+00 |-7.684065e-01 |
|  902  |22.5500 |  1   |   0.494571   |   0.674221   |  -5.352664   |-9.795983e+00 |-7.683562e-01 |
|  903  |22.5750 |  1   |   0.484502   |   0.666114   |  -5.348923   |-9.796658e+00 |-7.683051e-01 |
|  904  |22.6000 |  1   |   0.500552   |   0.659357   |  -5.345406   |-9.797343e+00 |-7.682532e-01 |
|  905  |22.6250 |  1   |   0.474210   |   0.675963   |  -5.341552   |-9.798036e+00 |-7.682004e-01 |
|  906  |22.6500 |  1   |   0.512181   |   0.642814   |  -5.337793   |-9.798737e+00 |-7.681468e-01 |
|  907  |22.6750 |  1   |   0.530902   |   0.671671   |  -5.334230   |-9.799447e+00 |-7.680925e-01 |
|  908  |22.7000 |  1   |   0.514472   |   0.642086   |  -5.330476   |-9.800166e+00 |-7.680373e-01 |
|  909  |22.7250 |  1   |   0.511914   |   0.644290   |  -5.326850   |-9.800893e+00 |-7.679813e-01 |
|  910  |22.7500 |  1   |   0.510064   |   0.648282   |  -5.323523   |-9.801628e+00 |-7.679245e-01 |
|  911  |22.7750 |  1   |   0.513836   |   0.638366   |  -5.319777   |-9.802371e+00 |-7.678670e-01 |
|  912  |22.8000 |  1   |   0.522687   |   0.642639   |  -5.316235   |-9.803123e+00 |-7.678087e-01 |
|  913  |22.8250 |  1   |   0.513428   |   0.637363   |  -5.312791   |-9.803882e+00 |-7.677496e-01 |
|  914  |22.8500 |  1   |   0.512795   |   0.639266   |  -5.309169   |-9.804650e+00 |-7.676897e-01 |
|  915  |22.8750 |  1   |   0.519954   |   0.638287   |  -5.305885   |-9.805425e+00 |-7.676291e-01 |
|  916  |22.9000 |  1   |   0.481317   |   0.673174   |  -5.302459   |-9.806209e+00 |-7.675677e-01 |
|  917  |22.9250 |  1   |   0.498218   |   0.656933   |  -5.299122   |-9.807000e+00 |-7.675056e-01 |
|  918  |22.9500 |  1   |   0.515562   |   0.638124   |  -5.295768   |-9.807799e+00 |-7.674427e-01 |
|  919  |22.9750 |  1   |   0.518832   |   0.657375   |  -5.292315   |-9.808606e+00 |-7.673791e-01 |
|  920  |23.0000 |  1   |   0.509751   |   0.646163   |  -5.289145   |-9.809420e+00 |-7.673148e-01 |
|  921  |23.0250 |  1   |   0.521466   |   0.646805   |  -5.285912   |-9.810242e+00 |-7.672497e-01 |
|  922  |23.0500 |  1   |   0.512008   |   0.637779   |  -5.282643   |-9.811071e+00 |-7.671840e-01 |
|  923  |23.0750 |  1   |   0.517762   |   0.647592   |  -5.279669   |-9.811908e+00 |-7.671175e-01 |
|  924  |23.1000 |  1   |   0.521314   |   0.644842   |  -5.276377   |-9.812752e+00 |-7.670503e-01 |
|  925  |23.1250 |  1   |   0.495290   |   0.652850   |  -5.273205   |-9.813603e+00 |-7.669824e-01 |
|  926  |23.1500 |  1   |   0.516843   |   0.640946   |  -5.270279   |-9.814461e+00 |-7.669139e-01 |
|  927  |23.1750 |  1   |   0.511137   |   0.637577   |  -5.267108   |-9.815327e+00 |-7.668446e-01 |
|  928  |23.2000 |  1   |   0.512721   |   0.636781   |  -5.264265   |-9.816199e+00 |-7.667747e-01 |
|  929  |23.2250 |  1   |   0.514486   |   0.641128   |  -5.261287   |-9.817078e+00 |-7.667042e-01 |
|  930  |23.2500 |  1   |   0.502929   |   0.654364   |  -5.258338   |-9.817965e+00 |-7.666329e-01 |
|  931  |23.2750 |  1   |   0.482228   |   0.677815   |  -5.255424   |-9.818858e+00 |-7.665610e-01 |
|  932  |23.3000 |  1   |   0.476847   |   0.674906   |  -5.252544   |-9.819758e+00 |-7.664885e-01 |
|  933  |23.3250 |  1   |   0.479599   |   0.666247   |  -5.249762   |-9.820664e+00 |-7.664153e-01 |
|  934  |23.3500 |  1   |   0.481180   |   0.669578   |  -5.247035   |-9.821577e+00 |-7.663414e-01 |
|  935  |23.3750 |  1   |   0.480612   |   0.667914   |  -5.244279   |-9.822497e+00 |-7.662670e-01 |
|  936  |23.4000 |  1   |   0.494679   |   0.679912   |  -5.241565   |-9.823423e+00 |-7.661919e-01 |
|  937  |23.4250 |  1   |   0.480638   |   0.670267   |  -5.238915   |-9.824355e+00 |-7.661162e-01 |
|  938  |23.4500 |  1   |   0.476561   |   0.677960   |  -5.236134   |-9.825294e+00 |-7.660399e-01 |
|  939  |23.4750 |  1   |   0.479632   |   0.672821   |  -5.233690   |-9.826238e+00 |-7.659630e-01 |
|  940  |23.5000 |  1   |   0.473291   |   0.676737   |  -5.231094   |-9.827189e+00 |-7.658855e-01 |
|  941  |23.5250 |  1   |   0.474442   |   0.677697   |  -5.228548   |-9.828146e+00 |-7.658074e-01 |
|  942  |23.5500 |  1   |   0.485998   |   0.658387   |  -5.226139   |-9.829110e+00 |-7.657288e-01 |
|  943  |23.5750 |  1   |   0.514283   |   0.634293   |  -5.223562   |-9.830079e+00 |-7.656495e-01 |
|  944  |23.6000 |  1   |   0.516713   |   0.646159   |  -5.221143   |-9.831053e+00 |-7.655697e-01 |
|  945  |23.6250 |  1   |   0.509222   |   0.641191   |  -5.218806   |-9.832034e+00 |-7.654894e-01 |
|  946  |23.6500 |  1   |   0.511243   |   0.639904   |  -5.216415   |-9.833020e+00 |-7.654084e-01 |
|  947  |23.6750 |  1   |   0.504852   |   0.645054   |  -5.214173   |-9.834012e+00 |-7.653270e-01 |
|  948  |23.7000 |  1   |   0.481507   |   0.668694   |  -5.211857   |-9.835010e+00 |-7.652450e-01 |
|  949  |23.7250 |  1   |   0.478430   |   0.667698   |  -5.209544   |-9.836013e+00 |-7.651624e-01 |
|  950  |23.7500 |  1   |   0.473767   |   0.672205   |  -5.207393   |-9.837021e+00 |-7.650793e-01 |
|  951  |23.7750 |  1   |   0.477168   |   0.673745   |  -5.205109   |-9.838035e+00 |-7.649957e-01 |
|  952  |23.8000 |  1   |   0.473230   |   0.681051   |  -5.203066   |-9.839054e+00 |-7.649116e-01 |
|  953  |23.8250 |  1   |   0.474712   |   0.669055   |  -5.200959   |-9.840078e+00 |-7.648270e-01 |
|  954  |23.8500 |  1   |   0.478994   |   0.672140   |  -5.198806   |-9.841107e+00 |-7.647419e-01 |
|  955  |23.8750 |  1   |   0.510193   |   0.643556   |  -5.196823   |-9.842141e+00 |-7.646563e-01 |
|  956  |23.9000 |  1   |   0.510342   |   0.649685   |  -5.194717   |-9.843180e+00 |-7.645702e-01 |
|  957  |23.9250 |  1   |   0.502699   |   0.645553   |  -5.192758   |-9.844224e+00 |-7.644836e-01 |
|  958  |23.9500 |  1   |   0.509565   |   0.643627   |  -5.190837   |-9.845273e+00 |-7.643966e-01 |
|  959  |23.9750 |  1   |   0.505842   |   0.650800   |  -5.188929   |-9.846327e+00 |-7.643091e-01 |
|  960  |24.0000 |  1   |   0.509080   |   0.639700   |  -5.187018   |-9.847385e+00 |-7.642211e-01 |
|  961  |24.0250 |  1   |   0.506969   |   0.643642   |  -5.185180   |-9.848448e+00 |-7.641327e-01 |
|  962  |24.0500 |  1   |   0.502202   |   0.644327   |  -5.183292   |-9.849515e+00 |-7.640438e-01 |
|  963  |24.0750 |  1   |   0.504608   |   0.646704   |  -5.181523   |-9.850587e+00 |-7.639545e-01 |
|  964  |24.1000 |  1   |   0.506113   |   0.640549   |  -5.179790   |-9.851663e+00 |-7.638648e-01 |
|  965  |24.1250 |  1   |   0.516273   |   0.650328   |  -5.178029   |-9.852743e+00 |-7.637746e-01 |
|  966  |24.1500 |  1   |   0.514624   |   0.656699   |  -5.176397   |-9.853828e+00 |-7.636840e-01 |
|  967  |24.1750 |  1   |   0.509777   |   0.652485   |  -5.174642   |-9.854916e+00 |-7.635931e-01 |
|  968  |24.2000 |  1   |   0.506223   |   0.644373   |  -5.173014   |-9.856009e+00 |-7.635017e-01 |
|  969  |24.2250 |  1   |   0.497977   |   0.651589   |  -5.171420   |-9.857106e+00 |-7.634099e-01 |
|  970  |24.2500 |  1   |   0.542505   |   0.823516   |  -5.169807   |-9.858206e+00 |-7.633177e-01 |
|  971  |24.2750 |  1   |   0.458664   |   0.726943   |  -5.168319   |-9.859311e+00 |-7.632251e-01 |
|  972  |24.3000 |  1   |   0.501798   |   0.644348   |  -5.166790   |-9.860419e+00 |-7.631322e-01 |
|  973  |24.3250 |  1   |   0.514156   |   0.663782   |  -5.165249   |-9.861531e+00 |-7.630389e-01 |
|  974  |24.3500 |  1   |   0.509705   |   0.649251   |  -5.163825   |-9.862646e+00 |-7.629452e-01 |
|  975  |24.3750 |  1   |   0.500570   |   0.646102   |  -5.162345   |-9.863765e+00 |-7.628512e-01 |
|  976  |24.4000 |  1   |   0.501039   |   0.644028   |  -5.160961   |-9.864888e+00 |-7.627568e-01 |
|  977  |24.4250 |  1   |   0.504128   |   0.646994   |  -5.159634   |-9.866014e+00 |-7.626621e-01 |
|  978  |24.4500 |  1   |   0.509705   |   0.643946   |  -5.158234   |-9.867143e+00 |-7.625670e-01 |
|  979  |24.4750 |  1   |   0.504889   |   0.642303   |  -5.156955   |-9.868275e+00 |-7.624717e-01 |
|  980  |24.5000 |  1   |   0.511998   |   0.654649   |  -5.155620   |-9.869411e+00 |-7.623759e-01 |
|  981  |24.5250 |  1   |   0.505531   |   0.643071   |  -5.154346   |-9.870549e+00 |-7.622799e-01 |
|  982  |24.5500 |  1   |   0.505855   |   0.644088   |  -5.153156   |-9.871691e+00 |-7.621836e-01 |
|  983  |24.5750 |  1   |   0.503710   |   0.643562   |  -5.151921   |-9.872835e+00 |-7.620870e-01 |
|  984  |24.6000 |  1   |   0.503625   |   0.652749   |  -5.150775   |-9.873983e+00 |-7.619900e-01 |
|  985  |24.6250 |  1   |   0.497902   |   0.668819   |  -5.149604   |-9.875133e+00 |-7.618928e-01 |
|  986  |24.6500 |  1   |   0.508365   |   0.655839   |  -5.148454   |-9.876286e+00 |-7.617953e-01 |
|  987  |24.6750 |  1   |   0.499875   |   0.649295   |  -5.147360   |-9.877441e+00 |-7.616975e-01 |
|  988  |24.7000 |  1   |   0.507928   |   0.651539   |  -5.146301   |-9.878599e+00 |-7.615995e-01 |
|  989  |24.7250 |  1   |   0.504901   |   0.643364   |  -5.145247   |-9.879760e+00 |-7.615012e-01 |
|  990  |24.7500 |  1   |   0.496468   |   0.650975   |  -5.144263   |-9.880923e+00 |-7.614026e-01 |
|  991  |24.7750 |  1   |   0.485625   |   0.660757   |  -5.143244   |-9.882088e+00 |-7.613038e-01 |
|  992  |24.8000 |  1   |   0.511090   |   0.655112   |  -5.142258   |-9.883256e+00 |-7.612048e-01 |
|  993  |24.8250 |  1   |   0.499859   |   0.652944   |  -5.141341   |-9.884426e+00 |-7.611055e-01 |
|  994  |24.8500 |  1   |   0.503584   |   0.657031   |  -5.140395   |-9.885598e+00 |-7.610060e-01 |
|  995  |24.8750 |  1   |   0.487252   |   0.669019   |  -5.139545   |-9.886772e+00 |-7.609062e-01 |
|  996  |24.9000 |  1   |   0.486750   |   0.677291   |  -5.138686   |-9.887948e+00 |-7.608063e-01 |
|  997  |24.9250 |  1   |   0.499252   |   0.655279   |  -5.137821   |-9.889126e+00 |-7.607061e-01 |
|  998  |24.9500 |  1   |   0.490417   |   0.702562   |  -5.137027   |-9.890305e+00 |-7.606058e-01 |
|  999  |24.9750 |  1   |   0.503546   |   0.651132   |  -5.136210   |-9.891487e+00 |-7.605052e-01 |
| 1000  |25.0000 |  1   |   0.483413   |   0.668655   |  -5.135456   |-9.892670e+00 |-7.604045e-01 |
| 1001  |25.0250 |  1   |   0.493134   |   0.660787   |  -5.134750   |-9.893855e+00 |-7.603035e-01 |
| 1002  |25.0500 |  1   |   0.499426   |   0.657750   |  -5.134009   |-9.895041e+00 |-7.602024e-01 |
| 1003  |25.0750 |  1   |   0.494080   |   0.661456   |  -5.133351   |-9.896229e+00 |-7.601011e-01 |
| 1004  |25.1000 |  1   |   0.507680   |   0.664709   |  -5.132659   |-9.897418e+00 |-7.599997e-01 |
| 1005  |25.1250 |  1   |   0.494671   |   0.669049   |  -5.132007   |-9.898609e+00 |-7.598981e-01 |
| 1006  |25.1500 |  1   |   0.509767   |   0.652732   |  -5.131421   |-9.899801e+00 |-7.597964e-01 |
| 1007  |25.1750 |  1   |   0.495492   |   0.653077   |  -5.130814   |-9.900994e+00 |-7.596945e-01 |
| 1008  |25.2000 |  1   |   0.498837   |   0.670956   |  -5.130269   |-9.902188e+00 |-7.595924e-01 |
| 1009  |25.2250 |  1   |   0.500010   |   0.651370   |  -5.129726   |-9.903383e+00 |-7.594903e-01 |
| 1010  |25.2500 |  1   |   0.499026   |   0.649779   |  -5.129186   |-9.904579e+00 |-7.593880e-01 |
| 1011  |25.2750 |  1   |   0.503929   |   0.657456   |  -5.128693   |-9.905776e+00 |-7.592856e-01 |
| 1012  |25.3000 |  1   |   0.502894   |   0.652709   |  -5.128228   |-9.906974e+00 |-7.591831e-01 |
| 1013  |25.3250 |  1   |   0.493295   |   0.659944   |  -5.127776   |-9.908173e+00 |-7.590805e-01 |
| 1014  |25.3500 |  1   |   0.496672   |   0.655548   |  -5.127374   |-9.909372e+00 |-7.589777e-01 |
| 1015  |25.3750 |  1   |   0.496958   |   0.668447   |  -5.126963   |-9.910572e+00 |-7.588749e-01 |
| 1016  |25.4000 |  1   |   0.477043   |   0.671800   |  -5.126575   |-9.911773e+00 |-7.587720e-01 |
| 1017  |25.4250 |  1   |   0.493859   |   0.660183   |  -5.126228   |-9.912974e+00 |-7.586690e-01 |
| 1018  |25.4500 |  1   |   0.497632   |   0.654123   |  -5.125876   |-9.914176e+00 |-7.585660e-01 |
| 1019  |25.4750 |  1   |   0.502519   |   0.653777   |  -5.125597   |-9.915378e+00 |-7.584628e-01 |
| 1020  |25.5000 |  1   |   0.499899   |   0.652161   |  -5.125308   |-9.916580e+00 |-7.583596e-01 |
| 1021  |25.5250 |  1   |   0.505590   |   0.660761   |  -5.125039   |-9.917782e+00 |-7.582564e-01 |
| 1022  |25.5500 |  1   |   0.499622   |   0.658219   |  -5.124808   |-9.918985e+00 |-7.581531e-01 |
| 1023  |25.5750 |  1   |   0.494296   |   0.658807   |  -5.124571   |-9.920187e+00 |-7.580498e-01 |
| 1024  |25.6000 |  1   |   0.499221   |   0.654806   |  -5.124375   |-9.921390e+00 |-7.579464e-01 |
| 1025  |25.6250 |  1   |   0.499540   |   0.652259   |  -5.124226   |-9.922592e+00 |-7.578430e-01 |
| 1026  |25.6500 |  1   |   0.497204   |   0.653026   |  -5.124062   |-9.923795e+00 |-7.577395e-01 |
| 1027  |25.6750 |  1   |   0.498324   |   0.660967   |  -5.123947   |-9.924997e+00 |-7.576361e-01 |
| 1028  |25.7000 |  1   |   0.495586   |   0.657316   |  -5.123838   |-9.926199e+00 |-7.575326e-01 |
| 1029  |25.7250 |  1   |   0.495336   |   0.659465   |  -5.123735   |-9.927401e+00 |-7.574291e-01 |
| 1030  |25.7500 |  1   |   0.498829   |   0.654207   |  -5.123691   |-9.928602e+00 |-7.573256e-01 |
| 1031  |25.7750 |  1   |   0.496229   |   0.656042   |  -5.123643   |-9.929803e+00 |-7.572221e-01 |
| 1032  |25.8000 |  1   |   0.498100   |   0.658060   |  -5.123639   |-9.931003e+00 |-7.571187e-01 |
| 1033  |25.8250 |  1   |   0.493844   |   0.656294   |  -5.123650   |-9.932203e+00 |-7.570152e-01 |
| 1034  |25.8500 |  1   |   0.496206   |   0.652885   |  -5.123656   |-9.933402e+00 |-7.569118e-01 |
| 1035  |25.8750 |  1   |   0.493092   |   0.655715   |  -5.123718   |-9.934600e+00 |-7.568084e-01 |
| 1036  |25.9000 |  1   |   0.499616   |   0.658027   |  -5.123774   |-9.935798e+00 |-7.567050e-01 |
| 1037  |25.9250 |  1   |   0.491226   |   0.658937   |  -5.123870   |-9.936995e+00 |-7.566016e-01 |
| 1038  |25.9500 |  1   |   0.500340   |   0.653876   |  -5.124001   |-9.938190e+00 |-7.564983e-01 |
| 1039  |25.9750 |  1   |   0.495677   |   0.650797   |  -5.124126   |-9.939385e+00 |-7.563951e-01 |
| 1040  |26.0000 |  1   |   0.476759   |   0.680443   |  -5.124280   |-9.940579e+00 |-7.562919e-01 |
| 1041  |26.0250 |  1   |   0.488234   |   0.659470   |  -5.124460   |-9.941772e+00 |-7.561888e-01 |
| 1042  |26.0500 |  1   |   0.504142   |   0.662927   |  -5.124649   |-9.942963e+00 |-7.560857e-01 |
| 1043  |26.0750 |  1   |   0.493124   |   0.657806   |  -5.124875   |-9.944154e+00 |-7.559827e-01 |
| 1044  |26.1000 |  1   |   0.494713   |   0.663481   |  -5.125130   |-9.945343e+00 |-7.558798e-01 |
| 1045  |26.1250 |  1   |   0.490307   |   0.656816   |  -5.125384   |-9.946530e+00 |-7.557770e-01 |
| 1046  |26.1500 |  1   |   0.484677   |   0.662911   |  -5.125675   |-9.947717e+00 |-7.556742e-01 |
| 1047  |26.1750 |  1   |   0.493045   |   0.654708   |  -5.125970   |-9.948901e+00 |-7.555716e-01 |
| 1048  |26.2000 |  1   |   0.538497   |   0.726460   |  -5.126294   |-9.950085e+00 |-7.554690e-01 |
| 1049  |26.2250 |  1   |   0.549513   |   0.737299   |  -5.126653   |-9.951266e+00 |-7.553666e-01 |
| 1050  |26.2500 |  1   |   0.492540   |   0.664606   |  -5.127014   |-9.952446e+00 |-7.552643e-01 |
| 1051  |26.2750 |  1   |   0.493577   |   0.659932   |  -5.127419   |-9.953625e+00 |-7.551620e-01 |
| 1052  |26.3000 |  1   |   0.467239   |   0.712753   |  -5.127825   |-9.954801e+00 |-7.550599e-01 |
| 1053  |26.3250 |  1   |   0.455075   |   0.725685   |  -5.128240   |-9.955976e+00 |-7.549580e-01 |
| 1054  |26.3500 |  1   |   0.475621   |   0.676184   |  -5.128711   |-9.957149e+00 |-7.548561e-01 |
| 1055  |26.3750 |  1   |   0.534025   |   0.705199   |  -5.129171   |-9.958320e+00 |-7.547544e-01 |
| 1056  |26.4000 |  1   |   0.535417   |   0.733416   |  -5.129674   |-9.959488e+00 |-7.546529e-01 |
| 1057  |26.4250 |  1   |   0.490559   |   0.667110   |  -5.130201   |-9.960655e+00 |-7.545515e-01 |
| 1058  |26.4500 |  1   |   0.492721   |   0.667776   |  -5.130724   |-9.961820e+00 |-7.544502e-01 |
| 1059  |26.4750 |  1   |   0.469686   |   0.716404   |  -5.131284   |-9.962982e+00 |-7.543491e-01 |
| 1060  |26.5000 |  1   |   0.461801   |   0.727427   |  -5.131858   |-9.964142e+00 |-7.542482e-01 |
| 1061  |26.5250 |  1   |   0.464589   |   0.694128   |  -5.132457   |-9.965300e+00 |-7.541474e-01 |
| 1062  |26.5500 |  1   |   0.471781   |   0.706331   |  -5.133085   |-9.966456e+00 |-7.540468e-01 |
| 1063  |26.5750 |  1   |   0.459214   |   0.698293   |  -5.133724   |-9.967609e+00 |-7.539464e-01 |
| 1064  |26.6000 |  1   |   0.465039   |   0.701240   |  -5.134387   |-9.968759e+00 |-7.538461e-01 |
| 1065  |26.6250 |  1   |   0.469236   |   0.698958   |  -5.135067   |-9.969907e+00 |-7.537461e-01 |
| 1066  |26.6500 |  1   |   0.464358   |   0.702767   |  -5.135759   |-9.971053e+00 |-7.536462e-01 |
| 1067  |26.6750 |  1   |   0.465766   |   0.699608   |  -5.136493   |-9.972195e+00 |-7.535466e-01 |
| 1068  |26.7000 |  1   |   0.459619   |   0.698973   |  -5.084913   |-9.973335e+00 |-7.534475e-01 |
| 1069  |26.7250 |  1   |   0.476216   |   0.700416   |  -5.145987   |-9.974473e+00 |-7.533490e-01 |
| 1070  |26.7500 |  1   |   0.480763   |   0.738047   |  -5.138887   |-9.975607e+00 |-7.532503e-01 |
| 1071  |26.7750 |  1   |   0.461171   |   0.701831   |  -5.135784   |-9.976739e+00 |-7.531516e-01 |
| 1072  |26.8000 |  1   |   0.447415   |   0.732048   |  -5.142042   |-9.977868e+00 |-7.530534e-01 |
| 1073  |26.8250 |  1   |   0.463453   |   0.697525   |  -5.138247   |-9.978994e+00 |-7.529551e-01 |
| 1074  |26.8500 |  1   |   0.461543   |   0.697473   |  -5.135863   |-9.980116e+00 |-7.528572e-01 |
| 1075  |26.8750 |  1   |   0.457196   |   0.716596   |  -5.141322   |-9.981236e+00 |-7.527593e-01 |
| 1076  |26.9000 |  1   |   0.468679   |   0.696296   |  -5.138910   |-9.982352e+00 |-7.526616e-01 |
| 1077  |26.9250 |  1   |   0.461713   |   0.706081   |  -5.146215   |-9.983465e+00 |-7.525640e-01 |
| 1078  |26.9500 |  1   |   0.466057   |   0.697697   |  -5.150453   |-9.984575e+00 |-7.524668e-01 |
| 1079  |26.9750 |  1   |   0.464665   |   0.702892   |  -5.143797   |-9.985682e+00 |-7.523698e-01 |
| 1080  |27.0000 |  1   |   0.460304   |   0.700288   |  -5.148337   |-9.986785e+00 |-7.522730e-01 |
| 1081  |27.0250 |  1   |   0.487071   |   0.704852   |  -5.146060   |-9.987885e+00 |-7.521765e-01 |
| 1082  |27.0500 |  1   |   0.491921   |   0.701263   |  -5.148301   |-9.988981e+00 |-7.520802e-01 |
| 1083  |27.0750 |  1   |   0.514891   |   0.691098   |  -5.155984   |-9.990075e+00 |-7.519842e-01 |
| 1084  |27.1000 |  1   |   0.511289   |   0.700004   |  -5.152389   |-9.991164e+00 |-7.518885e-01 |
| 1085  |27.1250 |  1   |   0.464907   |   0.689621   |  -5.154942   |-9.992250e+00 |-7.517932e-01 |
| 1086  |27.1500 |  1   |   0.522192   |   0.703924   |  -5.152321   |-9.993332e+00 |-7.516980e-01 |
| 1087  |27.1750 |  1   |   0.482099   |   0.686301   |  -5.150296   |-9.994411e+00 |-7.516032e-01 |
| 1088  |27.2000 |  1   |   0.484473   |   0.666338   |  -5.158286   |-9.995485e+00 |-7.515087e-01 |
| 1089  |27.2250 |  1   |   0.465606   |   0.714366   |  -5.158936   |-9.996557e+00 |-7.514145e-01 |
| 1090  |27.2500 |  1   |   0.455517   |   0.696506   |  -5.159762   |-9.997624e+00 |-7.513207e-01 |
| 1091  |27.2750 |  1   |   0.466815   |   0.703345   |  -5.162399   |-9.998687e+00 |-7.512272e-01 |
| 1092  |27.3000 |  1   |   0.456211   |   0.701992   |  -5.156860   |-9.999747e+00 |-7.511339e-01 |
| 1093  |27.3250 |  1   |   0.465226   |   0.696526   |  -5.159533   |-1.000080e+01 |-7.510410e-01 |
| 1094  |27.3500 |  1   |   0.512126   |   0.700777   |  -5.164709   |-1.000185e+01 |-7.509483e-01 |
| 1095  |27.3750 |  1   |   0.479834   |   0.671114   |  -5.164529   |-1.000290e+01 |-7.508561e-01 |
| 1096  |27.4000 |  1   |   0.492537   |   0.673946   |  -5.170199   |-1.000394e+01 |-7.507642e-01 |
| 1097  |27.4250 |  1   |   0.489561   |   0.672850   |  -5.167845   |-1.000498e+01 |-7.506725e-01 |
| 1098  |27.4500 |  1   |   0.488160   |   0.668952   |  -5.165967   |-1.000602e+01 |-7.505812e-01 |
| 1099  |27.4750 |  1   |   0.519246   |   0.695777   |  -5.169713   |-1.000705e+01 |-7.504902e-01 |
| 1100  |27.5000 |  1   |   0.489192   |   0.667250   |  -5.170115   |-1.000808e+01 |-7.503995e-01 |
| 1101  |27.5250 |  1   |   0.499167   |   0.682266   |  -5.175767   |-1.000910e+01 |-7.503093e-01 |
| 1102  |27.5500 |  1   |   0.492088   |   0.672850   |  -5.178382   |-1.001011e+01 |-7.502193e-01 |
| 1103  |27.5750 |  1   |   0.487768   |   0.663787   |  -5.175107   |-1.001113e+01 |-7.501297e-01 |
| 1104  |27.6000 |  1   |   0.493979   |   0.674322   |  -5.178327   |-1.001214e+01 |-7.500404e-01 |
| 1105  |27.6250 |  1   |   0.490692   |   0.663366   |  -5.177274   |-1.001314e+01 |-7.499515e-01 |
| 1106  |27.6500 |  1   |   0.492805   |   0.674422   |  -5.178603   |-1.001414e+01 |-7.498630e-01 |
| 1107  |27.6750 |  1   |   0.485858   |   0.664090   |  -5.187219   |-1.001513e+01 |-7.497748e-01 |
| 1108  |27.7000 |  1   |   0.486335   |   0.663156   |  -5.184418   |-1.001613e+01 |-7.496872e-01 |
| 1109  |27.7250 |  1   |   0.494404   |   0.702291   |  -5.186787   |-1.001711e+01 |-7.495997e-01 |
| 1110  |27.7500 |  1   |   0.446076   |   0.705973   |  -5.187599   |-1.001809e+01 |-7.495127e-01 |
| 1111  |27.7750 |  1   |   0.482750   |   0.662560   |  -5.184811   |-1.001907e+01 |-7.494261e-01 |
| 1112  |27.8000 |  1   |   0.476889   |   0.669748   |  -5.191664   |-1.002004e+01 |-7.493398e-01 |
| 1113  |27.8250 |  1   |   0.492741   |   0.670303   |  -5.192749   |-1.002101e+01 |-7.492541e-01 |
| 1114  |27.8500 |  1   |   0.481042   |   0.663581   |  -5.195669   |-1.002197e+01 |-7.491686e-01 |
| 1115  |27.8750 |  1   |   0.484773   |   0.664162   |  -5.197804   |-1.002292e+01 |-7.490836e-01 |
| 1116  |27.9000 |  1   |   0.471439   |   0.679314   |  -5.194899   |-1.002388e+01 |-7.489990e-01 |
| 1117  |27.9250 |  1   |   0.484387   |   0.664117   |  -5.198012   |-1.002482e+01 |-7.489147e-01 |
| 1118  |27.9500 |  1   |   0.485070   |   0.664848   |  -5.200681   |-1.002576e+01 |-7.488309e-01 |
| 1119  |27.9750 |  1   |   0.483672   |   0.663051   |  -5.202168   |-1.002670e+01 |-7.487475e-01 |
| 1120  |28.0000 |  1   |   0.483831   |   0.665529   |  -5.208302   |-1.002763e+01 |-7.486645e-01 |
| 1121  |28.0250 |  1   |   0.486482   |   0.662659   |  -5.207243   |-1.002856e+01 |-7.485818e-01 |
| 1122  |28.0500 |  1   |   0.484792   |   0.665026   |  -5.206583   |-1.002948e+01 |-7.484996e-01 |
| 1123  |28.0750 |  1   |   0.484451   |   0.669203   |  -5.210633   |-1.003040e+01 |-7.484179e-01 |
| 1124  |28.1000 |  1   |   0.483054   |   0.665843   |  -5.210285   |-1.003131e+01 |-7.483365e-01 |
| 1125  |28.1250 |  1   |   0.479900   |   0.666058   |  -5.215282   |-1.003221e+01 |-7.482556e-01 |
| 1126  |28.1500 |  1   |   0.481851   |   0.666201   |  -5.219299   |-1.003311e+01 |-7.481751e-01 |
| 1127  |28.1750 |  1   |   0.483100   |   0.663770   |  -5.218610   |-1.003401e+01 |-7.480950e-01 |
| 1128  |28.2000 |  1   |   0.485134   |   0.663042   |  -5.220768   |-1.003490e+01 |-7.480154e-01 |
| 1129  |28.2250 |  1   |   0.489655   |   0.667828   |  -5.221060   |-1.003578e+01 |-7.479362e-01 |
| 1130  |28.2500 |  1   |   0.480716   |   0.672463   |  -5.223165   |-1.003666e+01 |-7.478574e-01 |
| 1131  |28.2750 |  1   |   0.483455   |   0.666447   |  -5.228675   |-1.003753e+01 |-7.477791e-01 |
| 1132  |28.3000 |  1   |   0.483978   |   0.671011   |  -5.229529   |-1.003840e+01 |-7.477012e-01 |
| 1133  |28.3250 |  1   |   0.475710   |   0.669280   |  -5.232726   |-1.003926e+01 |-7.476238e-01 |
| 1134  |28.3500 |  1   |   0.479361   |   0.669395   |  -5.234044   |-1.004012e+01 |-7.475468e-01 |
| 1135  |28.3750 |  1   |   0.496386   |   0.682017   |  -5.232092   |-1.004097e+01 |-7.474703e-01 |
| 1136  |28.4000 |  1   |   0.485283   |   0.674847   |  -5.238405   |-1.004182e+01 |-7.473943e-01 |
| 1137  |28.4250 |  1   |   0.479666   |   0.672826   |  -5.240062   |-1.004266e+01 |-7.473187e-01 |
| 1138  |28.4500 |  1   |   0.479845   |   0.676287   |  -5.242591   |-1.004349e+01 |-7.472436e-01 |
| 1139  |28.4750 |  1   |   0.481158   |   0.666536   |  -5.247594   |-1.004432e+01 |-7.471689e-01 |
| 1140  |28.5000 |  1   |   0.483141   |   0.667308   |  -5.245477   |-1.004514e+01 |-7.470948e-01 |
| 1141  |28.5250 |  1   |   0.486443   |   0.666994   |  -5.248138   |-1.004596e+01 |-7.470210e-01 |
| 1142  |28.5500 |  1   |   0.479723   |   0.665961   |  -5.251076   |-1.004677e+01 |-7.469478e-01 |
| 1143  |28.5750 |  1   |   0.478889   |   0.668256   |  -5.253471   |-1.004757e+01 |-7.468750e-01 |
| 1144  |28.6000 |  1   |   0.480882   |   0.666810   |  -5.258529   |-1.004837e+01 |-7.468027e-01 |
| 1145  |28.6250 |  1   |   0.474088   |   0.675753   |  -5.259920   |-1.004917e+01 |-7.467310e-01 |
| 1146  |28.6500 |  1   |   0.480283   |   0.668231   |  -5.261212   |-1.004995e+01 |-7.466596e-01 |
| 1147  |28.6750 |  1   |   0.480788   |   0.674629   |  -5.263745   |-1.005073e+01 |-7.465887e-01 |
| 1148  |28.7000 |  1   |   0.480191   |   0.669986   |  -5.264326   |-1.005151e+01 |-7.465184e-01 |
| 1149  |28.7250 |  1   |   0.470319   |   0.675661   |  -5.269713   |-1.005228e+01 |-7.464485e-01 |
| 1150  |28.7500 |  1   |   0.480411   |   0.673238   |  -5.273651   |-1.005304e+01 |-7.463791e-01 |
| 1151  |28.7750 |  1   |   0.473258   |   0.678330   |  -5.274198   |-1.005380e+01 |-7.463102e-01 |
| 1152  |28.8000 |  1   |   0.478426   |   0.670399   |  -5.278360   |-1.005455e+01 |-7.462418e-01 |
| 1153  |28.8250 |  1   |   0.481835   |   0.677192   |  -5.278659   |-1.005530e+01 |-7.461739e-01 |
| 1154  |28.8500 |  1   |   0.477153   |   0.671792   |  -5.280378   |-1.005604e+01 |-7.461065e-01 |
| 1155  |28.8750 |  1   |   0.483710   |   0.675545   |  -5.286094   |-1.005677e+01 |-7.460396e-01 |
| 1156  |28.9000 |  1   |   0.478907   |   0.667440   |  -5.288677   |-1.005750e+01 |-7.459732e-01 |
| 1157  |28.9250 |  1   |   0.476032   |   0.683400   |  -5.291475   |-1.005822e+01 |-7.459074e-01 |
| 1158  |28.9500 |  1   |   0.478212   |   0.670153   |  -5.294549   |-1.005893e+01 |-7.458420e-01 |
| 1159  |28.9750 |  1   |   0.490696   |   0.674169   |  -5.294755   |-1.005964e+01 |-7.457771e-01 |
| 1160  |29.0000 |  1   |   0.478562   |   0.668860   |  -5.298482   |-1.006034e+01 |-7.457127e-01 |
| 1161  |29.0250 |  1   |   0.477472   |   0.670187   |  -5.301871   |-1.006104e+01 |-7.456489e-01 |
| 1162  |29.0500 |  1   |   0.481258   |   0.684804   |  -5.305298   |-1.006173e+01 |-7.455856e-01 |
| 1163  |29.0750 |  1   |   0.460059   |   0.686929   |  -5.310364   |-1.006241e+01 |-7.455228e-01 |
| 1164  |29.1000 |  1   |   0.448954   |   0.699926   |  -5.310048   |-1.006309e+01 |-7.454605e-01 |
| 1165  |29.1250 |  1   |   0.453478   |   0.709623   |  -5.313561   |-1.006376e+01 |-7.453987e-01 |
| 1166  |29.1500 |  1   |   0.445714   |   0.707670   |  -5.316738   |-1.006442e+01 |-7.453374e-01 |
| 1167  |29.1750 |  1   |   0.452987   |   0.698719   |  -5.318177   |-1.006508e+01 |-7.452767e-01 |
| 1168  |29.2000 |  1   |   0.449597   |   0.705269   |  -5.325022   |-1.006573e+01 |-7.452165e-01 |
| 1169  |29.2250 |  1   |   0.490915   |   0.676666   |  -5.327513   |-1.006637e+01 |-7.451568e-01 |
| 1170  |29.2500 |  1   |   0.449594   |   0.712068   |  -5.328948   |-1.006701e+01 |-7.450977e-01 |
| 1171  |29.2750 |  1   |   0.457821   |   0.707891   |  -5.332960   |-1.006764e+01 |-7.450391e-01 |
| 1172  |29.3000 |  1   |   0.451920   |   0.700959   |  -5.334478   |-1.006827e+01 |-7.449809e-01 |
| 1173  |29.3250 |  1   |   0.457635   |   0.705507   |  -5.338479   |-1.006889e+01 |-7.449234e-01 |
| 1174  |29.3500 |  1   |   0.447267   |   0.700000   |  -5.343494   |-1.006950e+01 |-7.448664e-01 |
| 1175  |29.3750 |  1   |   0.458476   |   0.705746   |  -5.346398   |-1.007011e+01 |-7.448099e-01 |
| 1176  |29.4000 |  1   |   0.450082   |   0.701366   |  -5.350024   |-1.007071e+01 |-7.447539e-01 |
| 1177  |29.4250 |  1   |   0.476268   |   0.681734   |  -5.351738   |-1.007130e+01 |-7.446985e-01 |
| 1178  |29.4500 |  1   |   0.460729   |   0.691428   |  -5.354504   |-1.007189e+01 |-7.446436e-01 |
| 1179  |29.4750 |  1   |   0.457731   |   0.717534   |  -5.359754   |-1.007247e+01 |-7.445893e-01 |
| 1180  |29.5000 |  1   |   0.448818   |   0.702547   |  -5.362285   |-1.007304e+01 |-7.445355e-01 |
| 1181  |29.5250 |  1   |   0.444427   |   0.710591   |  -5.367519   |-1.007361e+01 |-7.444822e-01 |
| 1182  |29.5500 |  1   |   0.449657   |   0.703114   |  -5.371200   |-1.007417e+01 |-7.444295e-01 |
| 1183  |29.5750 |  1   |   0.440618   |   0.711450   |  -5.371774   |-1.007472e+01 |-7.443773e-01 |
| 1184  |29.6000 |  1   |   0.446356   |   0.706056   |  -5.376509   |-1.007527e+01 |-7.443257e-01 |
| 1185  |29.6250 |  1   |   0.448475   |   0.708371   |  -5.380511   |-1.007581e+01 |-7.442746e-01 |
| 1186  |29.6500 |  1   |   0.448725   |   0.706641   |  -5.383604   |-1.007634e+01 |-7.442241e-01 |
| 1187  |29.6750 |  1   |   0.467714   |   0.685492   |  -5.389558   |-1.007687e+01 |-7.441741e-01 |
| 1188  |29.7000 |  1   |   0.469720   |   0.679917   |  -5.392343   |-1.007739e+01 |-7.441247e-01 |
| 1189  |29.7250 |  1   |   0.472234   |   0.676739   |  -5.394581   |-1.007790e+01 |-7.440758e-01 |
| 1190  |29.7500 |  1   |   0.471790   |   0.681953   |  -5.398936   |-1.007841e+01 |-7.440275e-01 |
| 1191  |29.7750 |  1   |   0.467655   |   0.680340   |  -5.401760   |-1.007891e+01 |-7.439797e-01 |
| 1192  |29.8000 |  1   |   0.474229   |   0.672687   |  -5.407802   |-1.007940e+01 |-7.439325e-01 |
| 1193  |29.8250 |  1   |   0.476449   |   0.675359   |  -5.411367   |-1.007989e+01 |-7.438858e-01 |
| 1194  |29.8500 |  1   |   0.476582   |   0.678527   |  -5.415002   |-1.008037e+01 |-7.438397e-01 |
| 1195  |29.8750 |  1   |   0.472185   |   0.686604   |  -5.419606   |-1.008084e+01 |-7.437942e-01 |
| 1196  |29.9000 |  1   |   0.471914   |   0.678537   |  -5.420797   |-1.008131e+01 |-7.437492e-01 |
| 1197  |29.9250 |  1   |   0.466368   |   0.687928   |  -5.426230   |-1.008177e+01 |-7.437047e-01 |
| 1198  |29.9500 |  1   |   0.471255   |   0.677494   |  -5.431831   |-1.008222e+01 |-7.436608e-01 |
| 1199  |29.9750 |  1   |   0.474277   |   0.679189   |  -5.434645   |-1.008267e+01 |-7.436175e-01 |
| 1200  |30.0000 |  1   |   0.470302   |   0.680280   |  -5.440101   |-1.008311e+01 |-7.435748e-01 |
...Finished
FINISHED - Elapsed time = 897.2004556 seconds
FINISHED - CPU process time = 1055.3392616 seconds

The resulting data structure that is returned from the call to main contains all the time-dependant variables for both the structural and aerodynamic solvers.

timestep_info can be found in case_data.structure and case_data.aero. It is an array with custom-made structure to contain the data of each solver.

In the .sharpy file, we can see which solvers are run:

flow = ['BeamLoader',
        'AerogridLoader',
        'StaticTrim',
        'BeamLoads',
        'AerogridPlot',
        'BeamPlot',
        'DynamicCoupled',
        ]

In order:

BeamLoader: reads the fem.h5 file and generates the structure for the beam solver.
AerogridLoader: reads the aero.h5 file and generates the aerodynamic grid for the aerodynamic solver.
StaticTrim: this solver performs a longitudinal trim (Thrust, Angle of attack and Elevator deflection) using the StaticCoupled solver.
BeamLoads: calculates the internal beam loads for the static solution
AerogridPlot: outputs the aerodynamic grid for the static solution.
BeamPlot: outputs the structural discretisation for the static solution.
DynamicCoupled: is the main driver of the dynamic simulation: executes the structural and aerodynamic solvers and couples both. Every converged time step is followed by a BeamLoads, AerogridPlot and BeamPlot execution.

Structural data organisation¶

The timestep_info structure contains several relevant variables:

for_pos: position of the body-attached frame of reference in inertial FoR.
for_vel: velocity (in body FoR) of the body FoR wrt inertial FoR.
pos: nodal position in A FoR.
psi: nodal rotations (from the material B FoR to A FoR) in a Cartesian Rotation Vector parametrisation.
applied_steady_forces: nodal forces from the aero solver and the applied forces.
postproc_cell: is a dictionary that contains the variables generated by a postprocessor, such as the internal beam loads.

The structural timestep_info also contains some useful variables:

cag and cga return $C^{AG}$ and $C^{GA}$, the rotation matrices from the body-attached (A) FoR to the inertial (G).
glob_pos rotates the pos variable to give you the inertial nodal position. If include_rbm = True is passed, for_pos is added to it.

Aerodynamic data organisation¶

The aerodynamic datastructure can be found in case_data.aero.timestep_info. It contains useful variables, such as:

dimensions and dimensions_star: gives the dimensions of every surface and wake surface. Organised as: dimensions[i_surf, 0] = chordwise panels, dimensions[i_surf, 1] = spanwise panels.
zeta and zeta_star: they are the $G$ FoR coordinates of the surface vertices.
gamma and gamma_star: vortex ring circulations.

Structural dynamics¶

We can now plot the rigid body dynamics:

RBM trajectory

[8]:

fig, ax = plt.subplots(1, 1, figsize=(7, 4))

# extract information
n_tsteps = len(case_data.structure.timestep_info)
xz = np.zeros((n_tsteps, 2))
for it in range(n_tsteps):
    xz[it, 0] = -case_data.structure.timestep_info[it].for_pos[0] # the - is so that increasing time -> increasing x
    xz[it, 1] = case_data.structure.timestep_info[it].for_pos[2]
ax.plot(xz[:, 0], xz[:, 1])
fig.suptitle('Longitudinal trajectory of the T-Tail model in a 20% 1-cos gust encounter')
ax.set_xlabel('X [m]')
ax.set_ylabel('Z [m]');
plt.show()

_images/content_example_notebooks_nonlinear_t-tail_HALE_22_0.svg

RBM velocities

[10]:

fig, ax = plt.subplots(3, 1, figsize=(7, 6), sharex=True)
ylabels = ['Vx [m/s]', 'Vy [m/s]', 'Vz [m/s]']

# extract information
n_tsteps = len(case_data.structure.timestep_info)
dt = case_data.settings['DynamicCoupled']['dt']
time_vec = np.linspace(0, n_tsteps*dt, n_tsteps)
for_vel = np.zeros((n_tsteps, 3))
for it in range(n_tsteps):
    for_vel[it, 0:3] = case_data.structure.timestep_info[it].for_vel[0:3]

for idim in range(3):
    ax[idim].plot(time_vec, for_vel[:, idim])
    ax[idim].set_ylabel(ylabels[idim])

ax[2].set_xlabel('time [s]')
plt.subplots_adjust(hspace=0)
fig.suptitle('Linear RBM velocities. T-Tail model in a 20% 1-cos gust encounter');
# ax.set_xlabel('X [m]')
# ax.set_ylabel('Z [m]');
plt.show()

_images/content_example_notebooks_nonlinear_t-tail_HALE_24_0.svg

[12]:

fig, ax = plt.subplots(3, 1, figsize=(7, 6), sharex=True)
ylabels = ['Roll rate [deg/s]', 'Pitch rate [deg/s]', 'Yaw rate [deg/s]']

# extract information
n_tsteps = len(case_data.structure.timestep_info)
dt = case_data.settings['DynamicCoupled']['dt']
time_vec = np.linspace(0, n_tsteps*dt, n_tsteps)
for_vel = np.zeros((n_tsteps, 3))
for it in range(n_tsteps):
    for_vel[it, 0:3] = case_data.structure.timestep_info[it].for_vel[3:6]*180/np.pi

for idim in range(3):
    ax[idim].plot(time_vec, for_vel[:, idim])
    ax[idim].set_ylabel(ylabels[idim])

ax[2].set_xlabel('time [s]')
plt.subplots_adjust(hspace=0)
fig.suptitle('Angular RBM velocities. T-Tail model in a 20% 1-cos gust encounter');
# ax.set_xlabel('X [m]')
# ax.set_ylabel('Z [m]');
plt.show()

_images/content_example_notebooks_nonlinear_t-tail_HALE_25_0.svg

Wing tip deformation

It is stored in timestep_info as pos. We need to find the correct node.

[13]:

fig, ax = plt.subplots(1, 1, figsize=(6, 6))
ax.scatter(case_data.structure.ini_info.pos[:, 0], case_data.structure.ini_info.pos[:, 1])
ax.axis('equal')
tip_node = np.argmax(case_data.structure.ini_info.pos[:, 1])
print('Wing tip node is the maximum Y one: ', tip_node)
ax.scatter(case_data.structure.ini_info.pos[tip_node, 0], case_data.structure.ini_info.pos[tip_node, 1], color='red')
plt.show()

Wing tip node is the maximum Y one:  16

_images/content_example_notebooks_nonlinear_t-tail_HALE_27_1.svg

We can plot now the pos[tip_node,:] variable:

[14]:

fig, ax = plt.subplots(1, 1, figsize=(7, 3))

# extract information
n_tsteps = len(case_data.structure.timestep_info)
xz = np.zeros((n_tsteps, 2))
for it in range(n_tsteps):
    xz[it, 0] = case_data.structure.timestep_info[it].pos[tip_node, 0]
    xz[it, 1] = case_data.structure.timestep_info[it].pos[tip_node, 2]
ax.plot(time_vec, xz[:, 1])
# fig.suptitle('Longitudinal trajectory of the T-Tail model in a 20% 1-cos gust encounter')
ax.set_xlabel('time [s]')
ax.set_ylabel('Vertical disp. [m]');
plt.show()

_images/content_example_notebooks_nonlinear_t-tail_HALE_29_0.svg

[15]:

fig, ax = plt.subplots(1, 1, figsize=(7, 3))

# extract information
n_tsteps = len(case_data.structure.timestep_info)
xz = np.zeros((n_tsteps, 2))
for it in range(n_tsteps):
    xz[it, 0] = case_data.structure.timestep_info[it].pos[tip_node, 0]
    xz[it, 1] = case_data.structure.timestep_info[it].pos[tip_node, 2]
ax.plot(time_vec, xz[:, 0])
# fig.suptitle('Longitudinal trajectory of the T-Tail model in a 20% 1-cos gust encounter')
ax.set_xlabel('time [s]')
ax.set_ylabel('Horizontal disp. [m]\nPositive is aft');
plt.show()

_images/content_example_notebooks_nonlinear_t-tail_HALE_30_0.svg

Wing root loads

The wing root loads can be extracted from the postproc_cell structure in timestep_info.

[17]:

fig, ax = plt.subplots(3, 1, figsize=(7, 6), sharex=True)
ylabels = ['Torsion [Nm2]', 'OOP [Nm2]', 'IP [Nm2]']

# extract information
n_tsteps = len(case_data.structure.timestep_info)
dt = case_data.settings['DynamicCoupled']['dt']
time_vec = np.linspace(0, n_tsteps*dt, n_tsteps)
loads = np.zeros((n_tsteps, 3))
for it in range(n_tsteps):
    loads[it, 0:3] = case_data.structure.timestep_info[it].postproc_cell['loads'][0, 3:6]

for idim in range(3):
    ax[idim].plot(time_vec, loads[:, idim])
    ax[idim].set_ylabel(ylabels[idim])

ax[2].set_xlabel('time [s]')
plt.subplots_adjust(hspace=0)
fig.suptitle('Wing root loads. T-Tail model in a 20% 1-cos gust encounter');
# ax.set_xlabel('X [m]')
# ax.set_ylabel('Z [m]');
plt.show()

_images/content_example_notebooks_nonlinear_t-tail_HALE_32_0.svg

Aerodynamic analysis¶

The aerodynamic analysis can be obviously conducted using python. However, the easiest way is to run the case by yourself and open the files in output/simple_HALE/beam and output/simple_HALE/aero with Paraview.

[18]:

url = 'https://raw.githubusercontent.com/ImperialCollegeLondon/sharpy/dev_doc/docs/source/content/example_notebooks/images/t-tail_solution.png'
Image(url=url, width=600)

[18]:

Asymptotic Stability of a Flying Wing in Cruise Trimmed Conditions¶

A Horten flying wing is analysed. The nonlinear trim condition is found and the system is linearised. The eigenvalues of the linearised system are then used to evaluate the stability at the cruise trimmed flight conditions.

[1]:

# required packages
import sharpy.utils.algebra as algebra
import sharpy.sharpy_main
from cases.hangar.richards_wing import Baseline
import numpy as np
import configobj
import matplotlib.pyplot as plt

Flight Conditions¶

Initial flight conditions. The values for angle of attack alpha, control surface deflection cs_deflection and thrust are only initial values. The values required for trim will be calculated by the StaticTrim routine

[2]:

u_inf = 28
alpha_deg = 4.5135
cs_deflection = 0.1814
thrust = 5.5129

Discretisation¶

[3]:

M = 4  # chordwise panels
N = 11  # spanwise panels
Msf = 5  # wake length in chord numbers

Create Horten Wing¶

[4]:

ws = Baseline(M=M,
              N=N,
              Mstarfactor=Msf,
              u_inf=u_inf,
              rho=1.02,
              alpha_deg=alpha_deg,
              roll_deg=0,
              cs_deflection_deg=cs_deflection,
              thrust=thrust,
              physical_time=20,
              case_name='horten',
              case_name_format=4,
              case_remarks='M%gN%gMsf%g' % (M, N, Msf))

ws.set_properties()
ws.initialise()
ws.clean_test_files()

ws.update_mass_stiffness(sigma=1., sigma_mass=2.5)
ws.update_fem_prop()
ws.generate_fem_file()
ws.update_aero_properties()
ws.generate_aero_file()

0
Section Mass: 11.88
Linear Mass: 11.88
Section Ixx: 1.8777
Section Iyy: 1.0137
Section Izz: 2.5496
Linear Ixx: 1.88
1
Section Mass: 10.99
Linear Mass: 10.99
Section Ixx: 1.4694
Section Iyy: 0.9345
Section Izz: 2.1501
Linear Ixx: 1.74
2
Section Mass: 10.10
Linear Mass: 10.10
Section Ixx: 1.1257
Section Iyy: 0.8561
Section Izz: 1.7993
Linear Ixx: 1.60
3
Section Mass: 9.21
Linear Mass: 9.21
Section Ixx: 0.8410
Section Iyy: 0.7783
Section Izz: 1.4933
Linear Ixx: 1.46
4
Section Mass: 8.32
Linear Mass: 8.32
Section Ixx: 0.6096
Section Iyy: 0.7011
Section Izz: 1.2280
Linear Ixx: 1.31
5
Section Mass: 7.43
Linear Mass: 7.43
Section Ixx: 0.4260
Section Iyy: 0.6246
Section Izz: 0.9996
Linear Ixx: 1.17
6
Section Mass: 6.54
Linear Mass: 6.54
Section Ixx: 0.2845
Section Iyy: 0.5485
Section Izz: 0.8040
Linear Ixx: 1.03
7
Section Mass: 5.64
Linear Mass: 5.64
Section Ixx: 0.1796
Section Iyy: 0.4728
Section Izz: 0.6374
Linear Ixx: 0.89
8
Section Mass: 4.75
Linear Mass: 4.75
Section Ixx: 0.1055
Section Iyy: 0.3975
Section Izz: 0.4959
Linear Ixx: 0.75
9
Section Mass: 3.86
Linear Mass: 3.86
Section Ixx: 0.0567
Section Iyy: 0.3226
Section Izz: 0.3753
Linear Ixx: 0.61
10
Section Mass: 2.97
Linear Mass: 2.97
Section Ixx: 0.0275
Section Iyy: 0.2479
Section Izz: 0.2719
Linear Ixx: 0.47

Simulation Information¶

The flow setting tells SHARPy which solvers to run and in which order. You may be stranged by the presence of the DynamicCoupled solver but it is necessary to give an initial speed to the structure. This will allow proper linearisation of the structural and rigid body equations.

[5]:

flow = ['BeamLoader',
        'AerogridLoader',
        'StaticTrim',
        'BeamPlot',
        'AerogridPlot',
        'AeroForcesCalculator',
        'DynamicCoupled',
        'Modal',
        'LinearAssembler',
        'AsymptoticStability',
        ]

SHARPy Settings¶

[6]:

settings = dict()
settings['SHARPy'] = {'case': ws.case_name,
                      'route': ws.case_route,
                      'flow': flow,
                      'write_screen': 'on',
                      'write_log': 'on',
                      'log_folder': './output/',
                      'log_file': ws.case_name + '.log'}

Loaders¶

[7]:

settings['BeamLoader'] = {'unsteady': 'off',
                          'orientation': algebra.euler2quat(np.array([ws.roll,
                                                                      ws.alpha,
                                                                      ws.beta]))}
settings['AerogridLoader'] = {'unsteady': 'off',
                              'aligned_grid': 'on',
                              'mstar': int(ws.M * ws.Mstarfactor),
                              'freestream_dir': ['1', '0', '0'],
                              'control_surface_deflection': [''],
                              'wake_shape_generator': 'StraightWake',
                              'wake_shape_generator_input': {'u_inf': ws.u_inf,
                                    'u_inf_direction': ['1', '0', '0'],
                                    'dt': ws.dt}}

StaticCoupled Solver¶

[8]:

settings['StaticCoupled'] = {'print_info': 'on',
                             'structural_solver': 'NonLinearStatic',
                             'structural_solver_settings': {'print_info': 'off',
                                                            'max_iterations': 200,
                                                            'num_load_steps': 1,
                                                            'delta_curved': 1e-5,
                                                            'min_delta': ws.tolerance,
                                                            'gravity_on': 'on',
                                                            'gravity': 9.81},
                             'aero_solver': 'StaticUvlm',
                             'aero_solver_settings': {'print_info': 'on',
                                                      'horseshoe': ws.horseshoe,
                                                      'num_cores': 4,
                                                      'n_rollup': int(0),
                                                      'rollup_dt': ws.dt,
                                                      'rollup_aic_refresh': 1,
                                                      'rollup_tolerance': 1e-4,
                                                      'velocity_field_generator': 'SteadyVelocityField',
                                                      'velocity_field_input': {'u_inf': ws.u_inf,
                                                                               'u_inf_direction': [1., 0, 0]},
                                                      'rho': ws.rho},
                             'max_iter': 200,
                             'n_load_steps': 1,
                             'tolerance': ws.tolerance,
                             'relaxation_factor': 0.2}

Trim solver¶

[9]:

settings['StaticTrim'] = {'solver': 'StaticCoupled',
                          'solver_settings': settings['StaticCoupled'],
                          'thrust_nodes': ws.thrust_nodes,
                          'initial_alpha': ws.alpha,
                          'initial_deflection': ws.cs_deflection,
                          'initial_thrust': ws.thrust,
                          'max_iter': 200,
                          'fz_tolerance': 1e-2,
                          'fx_tolerance': 1e-2,
                          'm_tolerance': 1e-2}

Nonlinear Equilibrium Post-process¶

[10]:

settings['AerogridPlot'] = {
                            'include_rbm': 'off',
                            'include_applied_forces': 'on',
                            'minus_m_star': 0,
                            'u_inf': ws.u_inf
                            }
settings['AeroForcesCalculator'] = {
                                    'write_text_file': 'off',
                                    'text_file_name': ws.case_name + '_aeroforces.csv',
                                    'screen_output': 'on',
                                    'unsteady': 'off',
                                    'coefficients': True,
                                    'q_ref': 0.5 * ws.rho * ws.u_inf ** 2,
                                    'S_ref': 12.809,
                                    }

settings['BeamPlot'] = {
                        'include_rbm': 'on',
                        'include_applied_forces': 'on',
                        'include_FoR': 'on'}

DynamicCoupled Solver¶

As mentioned before, a single time step of DynamicCoupled is required to give the structure the velocity required for the linearisation of the rigid body equations to be correct. Hence n_time_steps = 1

[11]:

struct_solver_settings = {'print_info': 'off',
                          'initial_velocity_direction': [-1., 0., 0.],
                          'max_iterations': 950,
                          'delta_curved': 1e-6,
                          'min_delta': ws.tolerance,
                          'newmark_damp': 5e-3,
                          'gravity_on': True,
                          'gravity': 9.81,
                          'num_steps': ws.n_tstep,
                          'dt': ws.dt,
                          'initial_velocity': ws.u_inf * 1}

step_uvlm_settings = {'print_info': 'on',
                      'num_cores': 4,
                      'convection_scheme': ws.wake_type,
                      'velocity_field_generator': 'SteadyVelocityField',
                      'velocity_field_input': {'u_inf': ws.u_inf * 0,
                                               'u_inf_direction': [1., 0., 0.]},
                      'rho': ws.rho,
                      'n_time_steps': ws.n_tstep,
                      'dt': ws.dt,
                      'gamma_dot_filtering': 3}

settings['DynamicCoupled'] = {'print_info': 'on',
                              'structural_solver': 'NonLinearDynamicCoupledStep',
                              'structural_solver_settings': struct_solver_settings,
                              'aero_solver': 'StepUvlm',
                              'aero_solver_settings': step_uvlm_settings,
                              'fsi_substeps': 200,
                              'fsi_tolerance': ws.fsi_tolerance,
                              'relaxation_factor': ws.relaxation_factor,
                              'minimum_steps': 1,
                              'relaxation_steps': 150,
                              'final_relaxation_factor': 0.5,
                              'n_time_steps': 1,
                              'dt': ws.dt,
                              'include_unsteady_force_contribution': 'off',
                                                          }

Linear Assembler Settings¶

Note that for the assembly of the linear system, we replace the parametrisation of the orientation with Euler angles instead of quaternions.

[13]:

settings['LinearAssembler'] = {'linear_system': 'LinearAeroelastic',
                               'linear_system_settings': {
                                   'beam_settings': {'modal_projection': 'off',
                                                     'inout_coords': 'modes',
                                                     'discrete_time': True,
                                                     'newmark_damp': 0.5e-2,
                                                     'discr_method': 'newmark',
                                                     'dt': ws.dt,
                                                     'proj_modes': 'undamped',
                                                     'num_modes': 9,
                                                     'print_info': 'on',
                                                     'gravity': 'on',
                                                     'remove_dofs': []},
                                   'aero_settings': {'dt': ws.dt,
                                                     'integr_order': 2,
                                                     'density': ws.rho,
                                                     'remove_predictor': 'off',
                                                     'use_sparse': 'off',
                                                     'remove_inputs': ['u_gust']},
                                   'track_body': 'on',
                                   'use_euler': 'on',
                                }}

Asymptotic Stability Post-processor¶

[14]:

settings['AsymptoticStability'] = {
                                    'print_info': 'on',
                                    'frequency_cutoff': 0,
                                    'export_eigenvalues': 'on',
                                    'num_evals': 100,
                                    }

Write solver file¶

[15]:

config = configobj.ConfigObj()
np.set_printoptions(precision=16)
file_name = ws.case_route + '/' + ws.case_name + '.sharpy'
config.filename = file_name
for k, v in settings.items():
    config[k] = v
config.write()

Run Simulation¶

[16]:

data = sharpy.sharpy_main.main(['', ws.case_route + '/' + ws.case_name + '.sharpy'])

--------------------------------------------------------------------------------
            ######  ##     ##    ###    ########  ########  ##    ##
           ##    ## ##     ##   ## ##   ##     ## ##     ##  ##  ##
           ##       ##     ##  ##   ##  ##     ## ##     ##   ####
            ######  ######### ##     ## ########  ########     ##
                 ## ##     ## ######### ##   ##   ##           ##
           ##    ## ##     ## ##     ## ##    ##  ##           ##
            ######  ##     ## ##     ## ##     ## ##           ##
--------------------------------------------------------------------------------
Aeroelastics Lab, Aeronautics Department.
    Copyright (c), Imperial College London.
    All rights reserved.
    License available at https://github.com/imperialcollegelondon/sharpy
Running SHARPy from /home/ng213/2TB/pazy_code/pazy-sharpy/lib/sharpy/docs/source/content/example_notebooks
SHARPy being run is in /home/ng213/2TB/pazy_code/pazy-sharpy/lib/sharpy
The branch being run is dev_setting_error
The version and commit hash are: v1.2.1-344-g0239644-0239644
SHARPy output folder set
     ./output//horten_u_inf2800_M4N11Msf5/
Generating an instance of BeamLoader
Variable for_pos has no assigned value in the settings file.
    will default to the value: [0.0, 0, 0]
Generating an instance of AerogridLoader
Variable control_surface_deflection_generator_settings has no assigned value in the settings file.
    will default to the value: {}
Variable dx1 has no assigned value in the settings file.
    will default to the value: -1.0
Variable ndx1 has no assigned value in the settings file.
    will default to the value: 1
Variable r has no assigned value in the settings file.
    will default to the value: 1.0
Variable dxmax has no assigned value in the settings file.
    will default to the value: -1.0
The aerodynamic grid contains 4 surfaces
  Surface 0, M=4, N=2
     Wake 0, M=20, N=2
  Surface 1, M=4, N=22
     Wake 1, M=20, N=22
  Surface 2, M=4, N=2
     Wake 2, M=20, N=2
  Surface 3, M=4, N=22
     Wake 3, M=20, N=22
  In total: 192 bound panels
  In total: 960 wake panels
  Total number of panels = 1152
Generating an instance of StaticTrim
Variable print_info has no assigned value in the settings file.
    will default to the value: True
Variable tail_cs_index has no assigned value in the settings file.
    will default to the value: 0
Variable initial_angle_eps has no assigned value in the settings file.
    will default to the value: 0.05
Variable initial_thrust_eps has no assigned value in the settings file.
    will default to the value: 2.0
Variable relaxation_factor has no assigned value in the settings file.
    will default to the value: 0.2
Variable save_info has no assigned value in the settings file.
    will default to the value: False
Generating an instance of StaticCoupled
Variable correct_forces_method has no assigned value in the settings file.
    will default to the value: 
Variable runtime_generators has no assigned value in the settings file.
    will default to the value: {}
Generating an instance of NonLinearStatic
Variable newmark_damp has no assigned value in the settings file.
    will default to the value: 0.0001
Variable gravity_dir has no assigned value in the settings file.
    will default to the value: [0.0, 0.0, 1.0]
Variable relaxation_factor has no assigned value in the settings file.
    will default to the value: 0.3
Variable dt has no assigned value in the settings file.
    will default to the value: 0.01
Variable num_steps has no assigned value in the settings file.
    will default to the value: 500
Variable initial_position has no assigned value in the settings file.
    will default to the value: [0. 0. 0.]
Generating an instance of StaticUvlm
Variable iterative_solver has no assigned value in the settings file.
    will default to the value: False
Variable iterative_tol has no assigned value in the settings file.
    will default to the value: 0.0001
Variable iterative_precond has no assigned value in the settings file.
    will default to the value: False
Variable cfl1 has no assigned value in the settings file.
    will default to the value: True
Variable vortex_radius has no assigned value in the settings file.
    will default to the value: 1e-06
Variable vortex_radius_wake_ind has no assigned value in the settings file.
    will default to the value: 1e-06
Variable rbm_vel_g has no assigned value in the settings file.
    will default to the value: [0.0, 0.0, 0.0, 0.0, 0.0, 0.0]
Variable centre_rot_g has no assigned value in the settings file.
    will default to the value: [0.0, 0.0, 0.0]



|=====|=====|============|==========|==========|==========|==========|==========|==========|
|iter |step | log10(res) |    Fx    |    Fy    |    Fz    |    Mx    |    My    |    Mz    |
|=====|=====|============|==========|==========|==========|==========|==========|==========|



|==========|==========|==========|==========|==========|==========|==========|==========|==========|==========|
|   iter   |alpha[deg]|elev[deg] |  thrust  |    Fx    |    Fy    |    Fz    |    Mx    |    My    |    Mz    |
|==========|==========|==========|==========|==========|==========|==========|==========|==========|==========|
|  0  |  0  |  0.00000   | -0.1051  | -0.0000  |  0.0598  | -0.0000  |  1.0837  | -0.0000  |
|  1  |  0  |  -7.62384  | -0.1284  | -0.0000  |  0.1276  | -0.0000  |  0.0045  | -0.0000  |
|  2  |  0  |  -8.33392  | -0.1190  | -0.0000  |  0.0397  | -0.0000  | -0.0774  | -0.0000  |
|  3  |  0  |  -9.30379  | -0.1133  |  0.0000  |  0.0011  | -0.0000  | -0.0070  |  0.0000  |
|  4  |  0  | -10.71602  | -0.1136  | -0.0000  |  0.0032  |  0.0000  | -0.0100  | -0.0000  |
|  5  |  0  | -10.88827  | -0.1138  | -0.0000  |  0.0043  |  0.0000  | -0.0119  | -0.0000  |
|  6  |  0  | -11.66331  | -0.1138  | -0.0000  |  0.0042  |  0.0000  | -0.0116  | -0.0000  |
|  7  |  0  | -12.88496  | -0.1138  | -0.0000  |  0.0041  |  0.0000  | -0.0116  |  0.0000  |
|    0     |  4.5135  |  0.1814  |  5.5129  | -0.1138  | -0.0000  |  0.0041  |  0.0000  | -0.0116  |  0.0000  |
|  0  |  0  |  0.00000   |-116.9870 | -0.0000  | 994.8063 | -0.0000  |-882.4104 | -0.0000  |
|  1  |  0  |  -5.79178  | -72.3841 |  0.0000  | 944.6140 |  0.0000  |-802.5912 |  0.0000  |
|  2  |  0  |  -6.63730  | -62.4378 |  0.0000  | 937.6662 |  0.0000  |-792.1259 | -0.0000  |
|  3  |  0  |  -7.22937  | -62.8923 | -0.0000  | 939.7866 | -0.0000  |-795.7093 | -0.0000  |
|  4  |  0  |  -8.65323  | -62.8757 | -0.0000  | 939.7100 | -0.0000  |-795.5764 | -0.0000  |
|  5  |  0  |  -8.81438  | -62.8640 | -0.0000  | 939.6554 | -0.0000  |-795.4837 | -0.0000  |
|  6  |  0  |  -9.59386  | -62.8660 |  0.0000  | 939.6644 | -0.0000  |-795.4991 |  0.0000  |
|  7  |  0  | -10.80422  | -62.8661 |  0.0000  | 939.6650 | -0.0000  |-795.5000 |  0.0000  |
|  8  |  0  | -11.01365  | -62.8660 | -0.0000  | 939.6647 | -0.0000  |-795.4994 | -0.0000  |
|  9  |  0  | -12.15197  | -62.8660 |  0.0000  | 939.6647 | -0.0000  |-795.4995 |  0.0000  |
|    0     |  7.3783  | -2.6834  |  5.5129  | -62.8660 |  0.0000  | 939.6647 | -0.0000  |-795.4995 |  0.0000  |
|  0  |  0  |  0.00000   | -32.9132 | -0.0000  | 371.4715 | -0.0000  |-902.7953 | -0.0000  |
|  1  |  0  |  -5.48409  | -8.7241  | -0.0000  | 298.8484 |  0.0000  |-777.2938 |  0.0000  |
|  2  |  0  |  -6.39387  | -4.0957  | -0.0000  | 289.8092 | -0.0000  |-761.4855 | -0.0000  |
|  3  |  0  |  -6.85613  | -4.6263  | -0.0000  | 293.2407 | -0.0000  |-767.3376 |  0.0000  |
|  4  |  0  |  -8.25962  | -4.6052  | -0.0000  | 293.1048 | -0.0000  |-767.1066 |  0.0000  |
|  5  |  0  |  -8.44065  | -4.5914  | -0.0000  | 293.0156 |  0.0000  |-766.9545 |  0.0000  |
|  6  |  0  |  -9.20968  | -4.5937  | -0.0000  | 293.0308 | -0.0000  |-766.9804 | -0.0000  |
|  7  |  0  | -10.44736  | -4.5939  | -0.0000  | 293.0316 | -0.0000  |-766.9819 | -0.0000  |
|  8  |  0  | -10.63000  | -4.5938  | -0.0000  | 293.0311 |  0.0000  |-766.9809 | -0.0000  |
|  9  |  0  | -11.74670  | -4.5938  |  0.0000  | 293.0311 |  0.0000  |-766.9810 |  0.0000  |
| 10  |  0  | -12.29943  | -4.5938  | -0.0000  | 293.0311 |  0.0000  |-766.9810 |  0.0000  |
|    0     |  4.5135  |  3.0462  |  5.5129  | -4.5938  | -0.0000  | 293.0311 |  0.0000  |-766.9810 |  0.0000  |
|  0  |  0  |  0.00000   | -4.1051  | -0.0000  |  0.0598  | -0.0000  |  1.0834  | -0.0000  |
|  1  |  0  |  -7.62384  | -4.1284  | -0.0000  |  0.1276  | -0.0000  |  0.0042  | -0.0000  |
|  2  |  0  |  -8.33392  | -4.1190  | -0.0000  |  0.0397  |  0.0000  | -0.0778  | -0.0000  |
|  3  |  0  |  -9.30379  | -4.1133  | -0.0000  |  0.0011  |  0.0000  | -0.0074  | -0.0000  |
|  4  |  0  | -10.71602  | -4.1136  |  0.0000  |  0.0032  | -0.0000  | -0.0104  |  0.0000  |
|  5  |  0  | -10.88827  | -4.1138  |  0.0000  |  0.0043  |  0.0000  | -0.0123  |  0.0000  |
|  6  |  0  | -11.66331  | -4.1138  | -0.0000  |  0.0042  |  0.0000  | -0.0119  | -0.0000  |
|  7  |  0  | -12.88496  | -4.1138  | -0.0000  |  0.0041  |  0.0000  | -0.0119  | -0.0000  |
|    0     |  4.5135  |  0.1814  |  7.5129  | -4.1138  | -0.0000  |  0.0041  |  0.0000  | -0.0119  | -0.0000  |
|  0  |  0  |  0.00000   |  0.0095  | -0.0000  |  0.0498  |  0.0000  |  1.1013  | -0.0000  |
|  1  |  0  |  -7.62357  | -0.0140  |  0.0000  |  0.1189  |  0.0000  |  0.0198  |  0.0000  |
|  2  |  0  |  -8.33375  | -0.0046  | -0.0000  |  0.0312  | -0.0000  | -0.0624  |  0.0000  |
|  3  |  0  |  -9.30318  |  0.0010  | -0.0000  | -0.0075  |  0.0000  |  0.0081  | -0.0000  |
|  4  |  0  | -10.71542  |  0.0007  | -0.0000  | -0.0054  |  0.0000  |  0.0051  |  0.0000  |
|  5  |  0  | -10.88766  |  0.0006  | -0.0000  | -0.0043  |  0.0000  |  0.0032  | -0.0000  |
|  6  |  0  | -11.66271  |  0.0006  |  0.0000  | -0.0044  | -0.0000  |  0.0035  |  0.0000  |
|  7  |  0  | -12.88441  |  0.0006  | -0.0000  | -0.0045  |  0.0000  |  0.0035  | -0.0000  |
|    1     |  4.5135  |  0.1814  |  5.4560  |  0.0006  | -0.0000  | -0.0045  |  0.0000  |  0.0035  | -0.0000  |
Generating an instance of BeamPlot
Variable include_applied_moments has no assigned value in the settings file.
    will default to the value: True
Variable name_prefix has no assigned value in the settings file.
    will default to the value: 
Variable output_rbm has no assigned value in the settings file.
    will default to the value: True
...Finished
Generating an instance of AerogridPlot
Variable include_forward_motion has no assigned value in the settings file.
    will default to the value: False
Variable include_unsteady_applied_forces has no assigned value in the settings file.
    will default to the value: False
Variable name_prefix has no assigned value in the settings file.
    will default to the value: 
Variable dt has no assigned value in the settings file.
    will default to the value: 0.0
Variable include_velocities has no assigned value in the settings file.
    will default to the value: False
Variable include_incidence_angle has no assigned value in the settings file.
    will default to the value: False
Variable num_cores has no assigned value in the settings file.
    will default to the value: 1
Variable vortex_radius has no assigned value in the settings file.
    will default to the value: 1e-06
...Finished
Generating an instance of AeroForcesCalculator
--------------------------------------------------------------------------------
tstep |   fx_g     |   fy_g     |   fz_g     |   Cfx_g    |   Cfy_g    |   Cfz_g   
    0 |  1.088e+01 | -4.476e-13 |  1.835e+03 |  2.124e-03 | -8.740e-17 |  3.583e-01
...Finished
Generating an instance of DynamicCoupled
Variable structural_substeps has no assigned value in the settings file.
    will default to the value: 0
Variable dynamic_relaxation has no assigned value in the settings file.
    will default to the value: False
Variable postprocessors has no assigned value in the settings file.
    will default to the value: []
Variable postprocessors_settings has no assigned value in the settings file.
    will default to the value: {}
Variable controller_id has no assigned value in the settings file.
    will default to the value: {}
Variable controller_settings has no assigned value in the settings file.
    will default to the value: {}
Variable cleanup_previous_solution has no assigned value in the settings file.
    will default to the value: False
Variable steps_without_unsteady_force has no assigned value in the settings file.
    will default to the value: 0
Variable pseudosteps_ramp_unsteady_force has no assigned value in the settings file.
    will default to the value: 0
Variable correct_forces_method has no assigned value in the settings file.
    will default to the value: 
Variable network_settings has no assigned value in the settings file.
    will default to the value: {}
Variable runtime_generators has no assigned value in the settings file.
    will default to the value: {}
Generating an instance of NonLinearDynamicCoupledStep
Variable num_load_steps has no assigned value in the settings file.
    will default to the value: 1
Variable gravity_dir has no assigned value in the settings file.
    will default to the value: [0.0, 0.0, 1.0]
Variable relaxation_factor has no assigned value in the settings file.
    will default to the value: 0.3
Variable balancing has no assigned value in the settings file.
    will default to the value: False
Generating an instance of StepUvlm
Variable iterative_solver has no assigned value in the settings file.
    will default to the value: False
Variable iterative_tol has no assigned value in the settings file.
    will default to the value: 0.0001
Variable iterative_precond has no assigned value in the settings file.
    will default to the value: False
Variable cfl1 has no assigned value in the settings file.
    will default to the value: True
Variable vortex_radius has no assigned value in the settings file.
    will default to the value: 1e-06
Variable vortex_radius_wake_ind has no assigned value in the settings file.
    will default to the value: 1e-06
Variable interp_coords has no assigned value in the settings file.
    will default to the value: 0
Variable filter_method has no assigned value in the settings file.
    will default to the value: 0
Variable interp_method has no assigned value in the settings file.
    will default to the value: 0
Variable yaw_slerp has no assigned value in the settings file.
    will default to the value: 0.0
Variable centre_rot has no assigned value in the settings file.
    will default to the value: [0.0, 0.0, 0.0]
Variable quasi_steady has no assigned value in the settings file.
    will default to the value: False



|=======|========|======|==============|==============|==============|==============|==============|
|  ts   |   t    | iter | struc ratio  |  iter time   | residual vel |  FoR_vel(x)  |  FoR_vel(z)  |
|=======|========|======|==============|==============|==============|==============|==============|

/home/ng213/2TB/pazy_code/pazy-sharpy/lib/sharpy/sharpy/aero/utils/uvlmlib.py:264: RuntimeWarning: invalid value encountered in true_divide
  flightconditions.uinf_direction = np.ctypeslib.as_ctypes(ts_info.u_ext[0][:, 0, 0]/flightconditions.uinf)

|   1   | 0.0089 |  3   |   0.877319   |   0.593232   |  -10.598250  |-2.791317e+01 |-2.203426e+00 |
...Finished
Generating an instance of Modal
Variable keep_linear_matrices has no assigned value in the settings file.
    will default to the value: True
Variable write_dat has no assigned value in the settings file.
    will default to the value: True
Variable delta_curved has no assigned value in the settings file.
    will default to the value: 0.01
Variable max_rotation_deg has no assigned value in the settings file.
    will default to the value: 15.0
Variable max_displacement has no assigned value in the settings file.
    will default to the value: 0.15
Variable use_custom_timestep has no assigned value in the settings file.
    will default to the value: -1
Structural eigenvalues



|==============|==============|==============|==============|==============|==============|==============|
|     mode     |  eval_real   |  eval_imag   | freq_n (Hz)  | freq_d (Hz)  |   damping    |  period (s)  |
|==============|==============|==============|==============|==============|==============|==============|

/home/ng213/2TB/pazy_code/pazy-sharpy/lib/sharpy/sharpy/solvers/modal.py:265: UserWarning: Projecting a system with damping on undamped modal shapes
  warnings.warn('Projecting a system with damping on undamped modal shapes')

|      0       |  -0.000000   |   0.000000   |   0.000000   |   0.000000   |   1.000000   |     inf      |
|      1       |   0.000000   |   0.000000   |   0.000000   |   0.000000   |   1.000000   |     inf      |
|      2       |   0.000000   |   0.000000   |   0.000000   |   0.000000   |   1.000000   |     inf      |
|      3       |   0.000000   |   0.000000   |   0.000000   |   0.000000   |   1.000000   |     inf      |
|      4       |  -0.000000   |   0.000000   |   0.000000   |   0.000000   |   1.000000   |     inf      |
|      5       |   0.000000   |   0.000000   |   0.000000   |   0.000000   |   1.000000   |     inf      |
|      6       |   0.000000   |   0.000000   |   0.000000   |   0.000000   |   1.000000   |     inf      |
|      7       |  -0.000000   |   0.000000   |   0.000000   |   0.000000   |   1.000000   |     inf      |
|      8       |  -0.000000   |   0.000000   |   0.000000   |   0.000000   |   1.000000   |     inf      |
|      9       |  -0.000000   |   0.000000   |   0.000000   |   0.000000   |   1.000000   |     inf      |
|      10      |   0.000000   |  28.293939   |   4.503120   |   4.503120   |  -0.000000   |   0.222068   |
|      11      |   0.000000   |  29.271318   |   4.658675   |   4.658675   |  -0.000000   |   0.214653   |
|      12      |   0.000000   |  54.780234   |   8.718545   |   8.718545   |  -0.000000   |   0.114698   |
|      13      |   0.000000   |  58.999779   |   9.390106   |   9.390106   |  -0.000000   |   0.106495   |
|      14      |   0.000000   |  70.520741   |  11.223724   |  11.223724   |  -0.000000   |   0.089097   |
|      15      |   0.000000   |  76.917111   |  12.241738   |  12.241738   |  -0.000000   |   0.081688   |
|      16      |   0.000000   |  87.324076   |  13.898058   |  13.898058   |  -0.000000   |   0.071952   |
|      17      |   0.000000   |  108.035577  |  17.194396   |  17.194396   |  -0.000000   |   0.058158   |
|      18      |   0.000000   |  119.692139  |  19.049596   |  19.049596   |  -0.000000   |   0.052495   |
|      19      |   0.000000   |  133.495187  |  21.246419   |  21.246419   |  -0.000000   |   0.047067   |
|      20      |   0.000000   |  134.444788  |  21.397553   |  21.397553   |  -0.000000   |   0.046734   |
|      21      |   0.000000   |  151.060442  |  24.042016   |  24.042016   |  -0.000000   |   0.041594   |
|      22      |   0.000000   |  159.369020  |  25.364367   |  25.364367   |  -0.000000   |   0.039425   |
|      23      |   0.000000   |  171.256102  |  27.256255   |  27.256255   |  -0.000000   |   0.036689   |
|      24      |   0.000000   |  173.895881  |  27.676389   |  27.676389   |  -0.000000   |   0.036132   |
|      25      |   0.000000   |  199.016557  |  31.674469   |  31.674469   |  -0.000000   |   0.031571   |
|      26      |   0.000000   |  205.412581  |  32.692428   |  32.692428   |  -0.000000   |   0.030588   |
|      27      |   0.000000   |  205.419531  |  32.693534   |  32.693534   |  -0.000000   |   0.030587   |
|      28      |   0.000000   |  223.563796  |  35.581283   |  35.581283   |  -0.000000   |   0.028105   |
|      29      |   0.000000   |  227.924750  |  36.275351   |  36.275351   |  -0.000000   |   0.027567   |
Generating an instance of LinearAssembler
Variable linearisation_tstep has no assigned value in the settings file.
    will default to the value: -1
Variable modal_tstep has no assigned value in the settings file.
    will default to the value: -1
Variable inout_coordinates has no assigned value in the settings file.
    will default to the value: 
Variable retain_inputs has no assigned value in the settings file.
    will default to the value: []
Variable retain_outputs has no assigned value in the settings file.
    will default to the value: []
Generating an instance of LinearAeroelastic
Variable uvlm_filename has no assigned value in the settings file.
    will default to the value: 
Generating an instance of LinearUVLM
Variable ScalingDict has no assigned value in the settings file.
    will default to the value: {}
Variable gust_assembler has no assigned value in the settings file.
    will default to the value: 
Variable rom_method has no assigned value in the settings file.
    will default to the value: []
Variable rom_method_settings has no assigned value in the settings file.
    will default to the value: {}
Variable vortex_radius has no assigned value in the settings file.
    will default to the value: 1e-06
Variable cfl1 has no assigned value in the settings file.
    will default to the value: True
Variable length has no assigned value in the settings file.
    will default to the value: 1.0
Variable speed has no assigned value in the settings file.
    will default to the value: 1.0
Variable density has no assigned value in the settings file.
    will default to the value: 1.0
Variable velocity_field_generator has no assigned value in the settings file.
    will default to the value: SteadyVelocityField
Variable velocity_field_input has no assigned value in the settings file.
    will default to the value: {}
Variable physical_model has no assigned value in the settings file.
    will default to the value: True
Variable track_body has no assigned value in the settings file.
    will default to the value: False
Variable track_body_number has no assigned value in the settings file.
    will default to the value: -1
Initialising Static linear UVLM solver class...
                        ...done in 0.39 sec
State-space realisation of UVLM equations started...
Computing wake propagation matrix with CFL1=True
     state-space model produced in form:
     x_{n+1} = A x_{n} + Bp u_{n+1}
                        ...done in 2.43 sec
Generating an instance of LinearBeam
Variable remove_sym_modes has no assigned value in the settings file.
    will default to the value: False
Warning, projecting system with damping onto undamped modes

Linearising gravity terms...
     M = 187.12 kg
     X_CG A -> 1.19 0.00 0.01
Node  1      -> B 0.000 -0.089 -0.000
                     -> A 0.089 0.206 0.000
                     -> G 0.089 0.206 -0.007
     Node mass:
             Matrix: 2.6141
Node  2      -> B -0.010 -0.019 -0.000
                     -> A 0.019 0.403 0.000
                     -> G 0.019 0.403 -0.001
     Node mass:
             Matrix: 7.3672
Node  3      -> B -0.019 -0.087 -0.000
                     -> A 0.234 0.800 0.000
                     -> G 0.234 0.800 -0.018
     Node mass:
             Matrix: 5.8780
Node  4      -> B -0.019 -0.084 -0.000
                     -> A 0.390 1.238 0.001
                     -> G 0.389 1.238 -0.030
     Node mass:
             Matrix: 2.8288
Node  5      -> B -0.018 -0.081 -0.000
                     -> A 0.546 1.676 0.001
                     -> G 0.544 1.676 -0.041
     Node mass:
             Matrix: 5.4372
Node  6      -> B -0.017 -0.078 -0.000
                     -> A 0.702 2.113 0.002
                     -> G 0.700 2.113 -0.053
     Node mass:
             Matrix: 2.6084
Node  7      -> B -0.016 -0.074 -0.000
                     -> A 0.857 2.551 0.003
                     -> G 0.855 2.551 -0.064
     Node mass:
             Matrix: 4.9963
Node  8      -> B -0.016 -0.071 -0.000
                     -> A 1.013 2.988 0.004
                     -> G 1.010 2.988 -0.076
     Node mass:
             Matrix: 2.3879
Node  9      -> B -0.015 -0.068 -0.000
                     -> A 1.169 3.426 0.005
                     -> G 1.166 3.426 -0.087
     Node mass:
             Matrix: 4.5555
Node 10      -> B -0.014 -0.065 -0.000
                     -> A 1.325 3.863 0.006
                     -> G 1.321 3.863 -0.098
     Node mass:
             Matrix: 2.1675
Node 11      -> B -0.013 -0.061 -0.000
                     -> A 1.480 4.301 0.007
                     -> G 1.476 4.301 -0.109
     Node mass:
             Matrix: 4.1146
Node 12      -> B -0.013 -0.058 -0.000
                     -> A 1.636 4.739 0.009
                     -> G 1.632 4.739 -0.120
     Node mass:
             Matrix: 1.9471
Node 13      -> B -0.012 -0.055 -0.000
                     -> A 1.792 5.176 0.010
                     -> G 1.787 5.176 -0.131
     Node mass:
             Matrix: 3.6738
Node 14      -> B -0.011 -0.052 -0.000
                     -> A 1.948 5.614 0.011
                     -> G 1.943 5.614 -0.142
     Node mass:
             Matrix: 1.7267
Node 15      -> B -0.011 -0.048 -0.000
                     -> A 2.104 6.052 0.012
                     -> G 2.098 6.052 -0.153
     Node mass:
             Matrix: 3.2329
Node 16      -> B -0.010 -0.045 -0.000
                     -> A 2.260 6.489 0.014
                     -> G 2.254 6.489 -0.164
     Node mass:
             Matrix: 1.5062
Node 17      -> B -0.009 -0.042 -0.000
                     -> A 2.415 6.927 0.015
                     -> G 2.409 6.927 -0.175
     Node mass:
             Matrix: 2.7921
Node 18      -> B -0.008 -0.039 -0.000
                     -> A 2.571 7.364 0.016
                     -> G 2.564 7.364 -0.186
     Node mass:
             Matrix: 1.2858
Node 19      -> B -0.008 -0.035 -0.000
                     -> A 2.727 7.802 0.017
                     -> G 2.720 7.802 -0.197
     Node mass:
             Matrix: 2.3512
Node 20      -> B -0.007 -0.032 -0.000
                     -> A 2.883 8.239 0.019
                     -> G 2.875 8.239 -0.208
     Node mass:
             Matrix: 1.0654
Node 21      -> B -0.006 -0.028 -0.000
                     -> A 3.038 8.677 0.020
                     -> G 3.030 8.677 -0.219
     Node mass:
             Matrix: 1.9104
Node 22      -> B -0.006 -0.026 -0.000
                     -> A 3.194 9.114 0.021
                     -> G 3.186 9.114 -0.230
     Node mass:
             Matrix: 0.8450
Node 23      -> B -0.005 -0.022 -0.000
                     -> A 3.350 9.552 0.023
                     -> G 3.341 9.552 -0.241
     Node mass:
             Matrix: 1.4695
Node 24      -> B -0.005 -0.022 -0.000
                     -> A 3.508 9.988 0.024
                     -> G 3.499 9.988 -0.252
     Node mass:
             Matrix: 0.3674
Node 25      -> B 0.000 0.089 -0.000
                     -> A 0.089 -0.206 0.000
                     -> G 0.089 -0.206 -0.007
     Node mass:
             Matrix: 2.6141
Node 26      -> B -0.010 0.019 -0.000
                     -> A 0.019 -0.403 0.000
                     -> G 0.019 -0.403 -0.001
     Node mass:
             Matrix: 7.3672
Node 27      -> B -0.019 0.087 -0.000
                     -> A 0.234 -0.800 0.000
                     -> G 0.234 -0.800 -0.018
     Node mass:
             Matrix: 5.8780
Node 28      -> B -0.019 0.084 -0.000
                     -> A 0.390 -1.238 0.001
                     -> G 0.389 -1.238 -0.030
     Node mass:
             Matrix: 2.8288
Node 29      -> B -0.018 0.081 -0.000
                     -> A 0.546 -1.676 0.001
                     -> G 0.544 -1.676 -0.041
     Node mass:
             Matrix: 5.4372
Node 30      -> B -0.017 0.078 -0.000
                     -> A 0.702 -2.113 0.002
                     -> G 0.700 -2.113 -0.053
     Node mass:
             Matrix: 2.6084
Node 31      -> B -0.016 0.074 -0.000
                     -> A 0.857 -2.551 0.003
                     -> G 0.855 -2.551 -0.064
     Node mass:
             Matrix: 4.9963
Node 32      -> B -0.016 0.071 -0.000
                     -> A 1.013 -2.988 0.004
                     -> G 1.010 -2.988 -0.076
     Node mass:
             Matrix: 2.3879
Node 33      -> B -0.015 0.068 -0.000
                     -> A 1.169 -3.426 0.005
                     -> G 1.166 -3.426 -0.087
     Node mass:
             Matrix: 4.5555
Node 34      -> B -0.014 0.065 -0.000
                     -> A 1.325 -3.863 0.006
                     -> G 1.321 -3.863 -0.098
     Node mass:
             Matrix: 2.1675
Node 35      -> B -0.013 0.061 -0.000
                     -> A 1.480 -4.301 0.007
                     -> G 1.476 -4.301 -0.109
     Node mass:
             Matrix: 4.1146
Node 36      -> B -0.013 0.058 -0.000
                     -> A 1.636 -4.739 0.009
                     -> G 1.632 -4.739 -0.120
     Node mass:
             Matrix: 1.9471
Node 37      -> B -0.012 0.055 -0.000
                     -> A 1.792 -5.176 0.010
                     -> G 1.787 -5.176 -0.131
     Node mass:
             Matrix: 3.6738
Node 38      -> B -0.011 0.052 -0.000
                     -> A 1.948 -5.614 0.011
                     -> G 1.943 -5.614 -0.142
     Node mass:
             Matrix: 1.7267
Node 39      -> B -0.011 0.048 -0.000
                     -> A 2.104 -6.052 0.012
                     -> G 2.098 -6.052 -0.153
     Node mass:
             Matrix: 3.2329
Node 40      -> B -0.010 0.045 -0.000
                     -> A 2.260 -6.489 0.014
                     -> G 2.254 -6.489 -0.164
     Node mass:
             Matrix: 1.5062
Node 41      -> B -0.009 0.042 -0.000
                     -> A 2.415 -6.927 0.015
                     -> G 2.409 -6.927 -0.175
     Node mass:
             Matrix: 2.7921
Node 42      -> B -0.008 0.039 -0.000
                     -> A 2.571 -7.364 0.016
                     -> G 2.564 -7.364 -0.186
     Node mass:
             Matrix: 1.2858
Node 43      -> B -0.008 0.035 -0.000
                     -> A 2.727 -7.802 0.017
                     -> G 2.720 -7.802 -0.197
     Node mass:
             Matrix: 2.3512
Node 44      -> B -0.007 0.032 -0.000
                     -> A 2.883 -8.239 0.019
                     -> G 2.875 -8.239 -0.208
     Node mass:
             Matrix: 1.0654
Node 45      -> B -0.006 0.028 -0.000
                     -> A 3.038 -8.677 0.020
                     -> G 3.030 -8.677 -0.219
     Node mass:
             Matrix: 1.9104
Node 46      -> B -0.006 0.026 -0.000
                     -> A 3.194 -9.114 0.021
                     -> G 3.186 -9.114 -0.230
     Node mass:
             Matrix: 0.8450
Node 47      -> B -0.005 0.022 -0.000
                     -> A 3.350 -9.552 0.023
                     -> G 3.341 -9.552 -0.241
     Node mass:
             Matrix: 1.4695
Node 48      -> B -0.005 0.022 -0.000
                     -> A 3.508 -9.988 0.024
                     -> G 3.499 -9.988 -0.252
     Node mass:
             Matrix: 0.3674
        Updated the beam C, modal C and K matrices with the terms from the
gravity linearisation

Aeroelastic system assembled:
     Aerodynamic states: 1536
     Structural states: 594
     Total states: 2130
     Inputs: 893
     Outputs: 891
Final system is:
State-space system
States: 2130
Inputs: 893
Outputs: 891

Generating an instance of AsymptoticStability
Variable reference_velocity has no assigned value in the settings file.
    will default to the value: 1.0
Variable display_root_locus has no assigned value in the settings file.
    will default to the value: False
Variable velocity_analysis has no assigned value in the settings file.
    will default to the value: []
Variable iterative_eigvals has no assigned value in the settings file.
    will default to the value: False
Variable modes_to_plot has no assigned value in the settings file.
    will default to the value: []
Variable postprocessors has no assigned value in the settings file.
    will default to the value: []
Variable postprocessors_settings has no assigned value in the settings file.
    will default to the value: {}
Dynamical System Eigenvalues
Calculating eigenvalues using direct method



|==============|==============|==============|==============|==============|==============|==============|
|     mode     |  eval_real   |  eval_imag   | freq_n (Hz)  | freq_d (Hz)  |   damping    |  period (s)  |
|==============|==============|==============|==============|==============|==============|==============|
|      0       |   0.000000   |   0.000000   |   0.000000   |   0.000000   |   1.000000   |     inf      |
|      1       |   0.000000   |   0.000000   |   0.000000   |   0.000000   |   1.000000   |     inf      |
|      2       |   0.000000   |   0.000000   |   0.000000   |   0.000000   |   1.000000   |     inf      |
|      3       |   0.000000   |   0.000000   |   0.000000   |   0.000000   |   1.000000   |     inf      |
|      4       |   0.000000   |   0.000000   |   0.000000   |   0.000000   |   1.000000   |     inf      |
|      5       |   0.000000   |   0.000000   |   0.000000   |   0.000000   |   1.000000   |     inf      |
|      6       |   0.000000   |   0.000000   |   0.000000   |   0.000000   |   1.000000   |     inf      |
|      7       |   0.000000   |   0.000000   |   0.000000   |   0.000000   |   1.000000   |     inf      |
|      8       |   0.000000   |   0.000000   |   0.000000   |   0.000000   |   1.000000   |     inf      |
|      9       |  -0.000000   |   0.000000   |   0.000000   |   0.000000   |   1.000000   |     inf      |
|      10      |  -0.000884   |  -0.321712   |   0.051202   |   0.051202   |   0.002747   |  19.530439   |
|      11      |  -0.000884   |   0.321712   |   0.051202   |   0.051202   |   0.002747   |  19.530439   |
|      12      |  -0.008470   |  -0.391287   |   0.062290   |   0.062275   |   0.021642   |  16.057731   |
|      13      |  -0.008470   |   0.391287   |   0.062290   |   0.062275   |   0.021642   |  16.057731   |
|      14      |  -0.022506   |   0.000000   |   0.003582   |   0.000000   |   1.000000   |     inf      |
|      15      |  -0.064808   |  -53.732514  |   8.551801   |   8.551795   |   0.001206   |   0.116935   |
|      16      |  -0.064808   |  53.732514   |   8.551801   |   8.551795   |   0.001206   |   0.116935   |
|      17      |  -0.101946   |  68.319141   |  10.873341   |  10.873329   |   0.001492   |   0.091968   |
|      18      |  -0.101946   |  -68.319141  |  10.873341   |  10.873329   |   0.001492   |   0.091968   |
|      19      |  -0.147587   |  83.265282   |  13.252102   |  13.252081   |   0.001772   |   0.075460   |
|      20      |  -0.147587   |  -83.265282  |  13.252102   |  13.252081   |   0.001772   |   0.075460   |
|      21      |  -0.248703   | -109.925782  |  17.495276   |  17.495232   |   0.002262   |   0.057158   |
|      22      |  -0.248703   |  109.925782  |  17.495276   |  17.495232   |   0.002262   |   0.057158   |
|      23      |  -0.293472   |  120.387631  |  19.160344   |  19.160287   |   0.002438   |   0.052191   |
|      24      |  -0.293472   | -120.387631  |  19.160344   |  19.160287   |   0.002438   |   0.052191   |
|      25      |  -0.350319   |  132.903280  |  21.152288   |  21.152214   |   0.002636   |   0.047276   |
|      26      |  -0.350319   | -132.903280  |  21.152288   |  21.152214   |   0.002636   |   0.047276   |
|      27      |  -0.376401   |  138.516954  |  22.045739   |  22.045658   |   0.002717   |   0.045360   |
|      28      |  -0.376401   | -138.516954  |  22.045739   |  22.045658   |   0.002717   |   0.045360   |
|      29      |  -0.494445   |  162.714232  |  25.896894   |  25.896774   |   0.003039   |   0.038615   |
|      30      |  -0.494445   | -162.714232  |  25.896894   |  25.896774   |   0.003039   |   0.038615   |
|      31      |  -0.511651   | -166.238306  |  26.457773   |  26.457648   |   0.003078   |   0.037796   |
|      32      |  -0.511651   |  166.238306  |  26.457773   |  26.457648   |   0.003078   |   0.037796   |
|      33      |  -0.559180   |  175.709816  |  27.965227   |  27.965086   |   0.003182   |   0.035759   |
|      34      |  -0.559180   | -175.709816  |  27.965227   |  27.965086   |   0.003182   |   0.035759   |
|      35      |  -0.569755   |  177.873274  |  28.309556   |  28.309411   |   0.003203   |   0.035324   |
|      36      |  -0.569755   | -177.873274  |  28.309556   |  28.309411   |   0.003203   |   0.035324   |
|      37      |  -0.669914   | -197.999020  |  31.512703   |  31.512523   |   0.003383   |   0.031733   |
|      38      |  -0.669914   |  197.999020  |  31.512703   |  31.512523   |   0.003383   |   0.031733   |
|      39      |  -0.678424   |  199.782714  |  31.796590   |  31.796406   |   0.003396   |   0.031450   |
|      40      |  -0.678424   | -199.782714  |  31.796590   |  31.796406   |   0.003396   |   0.031450   |
|      41      |  -0.715684   |  207.440562  |  33.015387   |  33.015191   |   0.003450   |   0.030289   |
|      42      |  -0.715684   | -207.440562  |  33.015387   |  33.015191   |   0.003450   |   0.030289   |
|      43      |  -0.721193   | -208.623018  |  33.203583   |  33.203385   |   0.003457   |   0.030117   |
|      44      |  -0.721193   |  208.623018  |  33.203583   |  33.203385   |   0.003457   |   0.030117   |
|      45      |  -0.796838   | -224.809928  |  35.779836   |  35.779611   |   0.003544   |   0.027949   |
|      46      |  -0.796838   |  224.809928  |  35.779836   |  35.779611   |   0.003544   |   0.027949   |
|      47      |  -0.801462   | -225.851223  |  35.945565   |  35.945339   |   0.003549   |   0.027820   |
|      48      |  -0.801462   |  225.851223  |  35.945565   |  35.945339   |   0.003549   |   0.027820   |
|      49      |  -0.823218   |  257.880058  |  41.043095   |  41.042886   |   0.003192   |   0.024365   |
|      50      |  -0.823218   | -257.880058  |  41.043095   |  41.042886   |   0.003192   |   0.024365   |
|      51      |  -0.829849   |  232.223377  |  36.959734   |  36.959498   |   0.003573   |   0.027057   |
|      52      |  -0.829849   | -232.223377  |  36.959734   |  36.959498   |   0.003573   |   0.027057   |
|      53      |  -0.833132   |  232.986723  |  37.081226   |  37.080989   |   0.003576   |   0.026968   |
|      54      |  -0.833132   | -232.986723  |  37.081226   |  37.080989   |   0.003576   |   0.026968   |
|      55      |  -0.837692   |  252.830750  |  40.239484   |  40.239264   |   0.003313   |   0.024851   |
|      56      |  -0.837692   | -252.830750  |  40.239484   |  40.239264   |   0.003313   |   0.024851   |
|      57      |  -0.843057   | -274.636584  |  43.709976   |  43.709770   |   0.003070   |   0.022878   |
|      58      |  -0.843057   |  274.636584  |  43.709976   |  43.709770   |   0.003070   |   0.022878   |
|      59      |  -0.855990   | -264.468496  |  42.091689   |  42.091468   |   0.003237   |   0.023758   |
|      60      |  -0.855990   |  264.468496  |  42.091689   |  42.091468   |   0.003237   |   0.023758   |
|      61      |  -0.864725   |  271.184095  |  43.160509   |  43.160289   |   0.003189   |   0.023169   |
|      62      |  -0.864725   | -271.184095  |  43.160509   |  43.160289   |   0.003189   |   0.023169   |
|      63      |  -0.871325   | -283.421756  |  45.108187   |  45.107973   |   0.003074   |   0.022169   |
|      64      |  -0.871325   |  283.421756  |  45.108187   |  45.107973   |   0.003074   |   0.022169   |
|      65      |  -0.878445   |  267.336890  |  42.548217   |  42.547987   |   0.003286   |   0.023503   |
|      66      |  -0.878445   | -267.336890  |  42.548217   |  42.547987   |   0.003286   |   0.023503   |
|      67      |  -0.882869   | -280.833495  |  44.696260   |  44.696039   |   0.003144   |   0.022373   |
|      68      |  -0.882869   |  280.833495  |  44.696260   |  44.696039   |   0.003144   |   0.022373   |
|      69      |  -0.884024   | -245.027542  |  38.997598   |  38.997344   |   0.003608   |   0.025643   |
|      70      |  -0.884024   |  245.027542  |  38.997598   |  38.997344   |   0.003608   |   0.025643   |
|      71      |  -0.886589   | -245.661879  |  39.098557   |  39.098302   |   0.003609   |   0.025577   |
|      72      |  -0.886589   |  245.661879  |  39.098557   |  39.098302   |   0.003609   |   0.025577   |
|      73      |  -0.891210   | -288.915188  |  45.982499   |  45.982280   |   0.003085   |   0.021748   |
|      74      |  -0.891210   |  288.915188  |  45.982499   |  45.982280   |   0.003085   |   0.021748   |
|      75      |  -0.908699   |  251.206723  |  39.981053   |  39.980792   |   0.003617   |   0.025012   |
|      76      |  -0.908699   | -251.206723  |  39.981053   |  39.980792   |   0.003617   |   0.025012   |
|      77      |  -0.910251   | -251.606127  |  40.044621   |  40.044359   |   0.003618   |   0.024972   |
|      78      |  -0.910251   |  251.606127  |  40.044621   |  40.044359   |   0.003618   |   0.024972   |
|      79      |  -0.914184   | -241.156682  |  38.381554   |  38.381278   |   0.003791   |   0.026054   |
|      80      |  -0.914184   |  241.156682  |  38.381554   |  38.381278   |   0.003791   |   0.026054   |
|      81      |  -0.915395   |  290.517030  |  46.237451   |  46.237221   |   0.003151   |   0.021628   |
|      82      |  -0.915395   | -290.517030  |  46.237451   |  46.237221   |   0.003151   |   0.021628   |
|      83      |  -0.933974   | -278.955360  |  44.397373   |  44.397124   |   0.003348   |   0.022524   |
|      84      |  -0.933974   |  278.955360  |  44.397373   |  44.397124   |   0.003348   |   0.022524   |
|      85      |  -0.943144   | -260.320871  |  41.431625   |  41.431353   |   0.003623   |   0.024136   |
|      86      |  -0.943144   |  260.320871  |  41.431625   |  41.431353   |   0.003623   |   0.024136   |
|      87      |  -0.944542   | -260.700481  |  41.492043   |  41.491770   |   0.003623   |   0.024101   |
|      88      |  -0.944542   |  260.700481  |  41.492043   |  41.491770   |   0.003623   |   0.024101   |
|      89      |  -0.953003   | -294.814043  |  46.921357   |  46.921112   |   0.003233   |   0.021312   |
|      90      |  -0.953003   |  294.814043  |  46.921357   |  46.921112   |   0.003233   |   0.021312   |
|      91      |  -0.960626   | -295.652742  |  47.054844   |  47.054595   |   0.003249   |   0.021252   |
|      92      |  -0.960626   |  295.652742  |  47.054844   |  47.054595   |   0.003249   |   0.021252   |
|      93      |  -0.960976   | -265.315963  |  42.226624   |  42.226347   |   0.003622   |   0.023682   |
|      94      |  -0.960976   |  265.315963  |  42.226624   |  42.226347   |   0.003622   |   0.023682   |
|      95      |  -0.961739   |  300.017779  |  47.749558   |  47.749313   |   0.003206   |   0.020943   |
|      96      |  -0.961739   | -300.017779  |  47.749558   |  47.749313   |   0.003206   |   0.020943   |
|      97      |  -0.961940   | -265.596058  |  42.271203   |  42.270925   |   0.003622   |   0.023657   |
|      98      |  -0.961940   |  265.596058  |  42.271203   |  42.270925   |   0.003622   |   0.023657   |
|      99      |  -0.965385   | -266.582863  |  42.428259   |  42.427980   |   0.003621   |   0.023569   |
FINISHED - Elapsed time = 16.3462206 seconds
FINISHED - CPU process time = 68.0131146 seconds

Post-processing¶

Nonlinear Equilibrium¶

The files can be opened with Paraview to see the deformation and aerodynamic loading on the flying wing in trim conditions.

Asymptotic Stability¶

[17]:

eigenvalues_trim = np.loadtxt('./output/horten_u_inf2800_M4N11Msf5/stability/eigenvalues.dat')

Flight Dynamics modes¶

The flight dynamics modes can be found close to the origin of the Argand diagram. In particular, the phugoid is the mode that is closest to the imaginary axis. An exercise is left to the user to compare this phugoid predicition with the nonlinear response!

[18]:

fig = plt.figure()
plt.scatter(eigenvalues_trim[:, 0], eigenvalues_trim[:, 1],
           marker='x',
           color='k')
plt.xlim(-0.5, 0.5)
plt.ylim(-0.5, 0.5)
plt.grid()
plt.xlabel('Real Part, $Re(\lambda)$ [rad/s]')
plt.ylabel('Imaginary Part, $Im(\lambda)$ [rad/s]');

_images/content_example_notebooks_linear_horten_37_0.png

Structural Modes¶

Looking further out on the plot, the structural modes appear. There is a curve given the Newmark-$\beta$ integration scheme and on top of it several modes are damped by the presence of the aerodynamics.

Try changing newmark_damp in the LinearAssembler settings to see how this plot changes!

[19]:

fig = plt.figure()
plt.scatter(eigenvalues_trim[:, 0], eigenvalues_trim[:, 1],
           marker='x',
           color='k')
plt.xlim(-5, 0.5)
plt.ylim(-200, 200)
plt.grid()
plt.xlabel('Real Part, $Re(\lambda)$ [rad/s]')
plt.ylabel('Imaginary Part, $Im(\lambda)$ [rad/s]');

_images/content_example_notebooks_linear_horten_39_0.png

Simulation NREL 5MW wind turbine¶

[1]:

%config InlineBackend.figure_format = 'svg'
from IPython.display import Image
url = 'https://raw.githubusercontent.com/ImperialCollegeLondon/sharpy/dev_doc/docs/source/content/example_notebooks/images/turbulence_no_legend.png'
Image(url=url, width=800)

[1]:

In this notebook the blade loads on the NREL-5MW reference wind turbine computed with SHARPy and OpenFAST will be compared. However, zero-drag airfoils have been used.

OpenFAST: https://openfast.readthedocs.io

NREL-5MW: Jonkman, J.; Butterfield, S.; Musial, W. and Scott, G.. Definition of a 5-MW Reference Wind Turbine for Offshore System Development, Technical Report, NREL 2009

Load the required packages:

[2]:

# Required packages
%matplotlib inline
import numpy as np
import os
import matplotlib.pyplot as plt

# Required SHARPy modules
import sharpy.sharpy_main
import sharpy.utils.algebra as algebra
import sharpy.utils.generate_cases as gc
import cases.templates.template_wt as template_wt

These are the results from the OpenFAST simulation for comparison: out-of-plane of_cNdrR and in-plane of_cTdrR coefficients along the blade and thrust of_ct and power of_cp rotor coefficients

[3]:

of_rR = np.array([0.20158356, 0.3127131, 0.40794048, 0.5984148, 0.6936519, 0.85238045, 0.899999, 0.95555407, 0.98729974, 1.0])
of_cNdrR = np.array([0.08621394, 0.14687876, 0.19345148, 0.2942731, 0.36003628, 0.43748564, 0.44762507, 0.38839236, 0.29782477, 0.0])
of_cTdrR = np.array([0.048268348, 0.051957503, 0.05304592, 0.052862607, 0.056001827, 0.0536646, 0.050112925, 0.038993906, 0.023664437, 0.0])

of_ct = 0.69787693
of_cp = 0.48813498

Create SHARPy case¶

Definition of parameters

[4]:

# Mathematical constants
deg2rad = np.pi/180.

# Case
case = 'rotor'
route = './'

# Geometry discretization
chord_panels = np.array([8], dtype=int)
revs_in_wake = 5

# Operation
rotation_velocity = 12.1*2*np.pi/60
pitch_deg = 0. #degrees

# Wind
WSP = 12.
air_density = 1.225

# Simulation
dphi = 4.*deg2rad
revs_to_simulate = 5

Computation of associated parameters

[5]:

dt = dphi/rotation_velocity
time_steps = int(revs_to_simulate*2.*np.pi/dphi)
mstar = int(revs_in_wake*2.*np.pi/dphi)

Generation of the rotor geometry based on the information in the excel file

[6]:

op_params = {'rotation_velocity': rotation_velocity,
             'pitch_deg': pitch_deg,
             'wsp': WSP,
             'dt': dt}

geom_params = {'chord_panels':chord_panels,
               'tol_remove_points': 1e-8,
               'n_points_camber': 100,
               'm_distribution': 'uniform'}

excel_description = {'excel_file_name': route + '/source/type02_db_NREL5MW_v02.xlsx',
                     'excel_sheet_parameters': 'parameters',
                     'excel_sheet_structural_blade': 'structural_blade',
                     'excel_sheet_discretization_blade': 'discretization_blade',
                     'excel_sheet_aero_blade': 'aero_blade',
                     'excel_sheet_airfoil_info': 'airfoil_info',
                     'excel_sheet_airfoil_chord': 'airfoil_coord'}

options = {'camber_effect_on_twist': False,
           'user_defined_m_distribution_type': None,
           'include_polars': False}

rotor = template_wt.rotor_from_excel_type03(op_params,
                                            geom_params,
                                            excel_description,
                                            options)

WARNING: The poisson cofficient is assumed equal to 0.3
WARNING: Cross-section area is used as shear area
WARNING: Using perpendicular axis theorem to compute the inertia around xB
WARNING: Replacing node 29 by node 0
WARNING: Replacing node 58 by node 0

Define simulation details. The steady simulation is faster than the dynamic simulation. However, the dynamic simulation includes wake self-induction and provides more accurate results.

[7]:

steady_simulation = False

[8]:

SimInfo = gc.SimulationInformation()
SimInfo.set_default_values()

if steady_simulation:
    SimInfo.solvers['SHARPy']['flow'] = ['BeamLoader',
                            'AerogridLoader',
                            'StaticCoupledRBM',
                            'BeamPlot',
                            'AerogridPlot',
                            'SaveData']
else:
    SimInfo.solvers['SHARPy']['flow'] = ['BeamLoader',
                            'AerogridLoader',
                            'StaticCoupledRBM',
                            'DynamicCoupled']

SimInfo.solvers['SHARPy']['case'] = case
SimInfo.solvers['SHARPy']['route'] = route
SimInfo.solvers['SHARPy']['write_log'] = True
SimInfo.set_variable_all_dicts('dt', dt)
SimInfo.set_variable_all_dicts('rho', air_density)

SimInfo.solvers['SteadyVelocityField']['u_inf'] = WSP
SimInfo.solvers['SteadyVelocityField']['u_inf_direction'] = np.array([0., 0., 1.])

SimInfo.solvers['BeamLoader']['unsteady'] = 'on'

SimInfo.solvers['AerogridLoader']['unsteady'] = 'on'
SimInfo.solvers['AerogridLoader']['mstar'] = mstar
SimInfo.solvers['AerogridLoader']['freestream_dir'] = np.array([0.,0.,0.])
SimInfo.solvers['AerogridLoader']['wake_shape_generator'] = 'HelicoidalWake'
SimInfo.solvers['AerogridLoader']['wake_shape_generator_input'] = {'u_inf': WSP,
                                                                   'u_inf_direction': SimInfo.solvers['SteadyVelocityField']['u_inf_direction'],
                                                                   'rotation_velocity': rotation_velocity*np.array([0., 0., 1.]),
                                                                   'dt': dt,
                                                                   'dphi1': dphi,
                                                                   'ndphi1': mstar,
                                                                   'r': 1.,
                                                                   'dphimax': 10*deg2rad}

SimInfo.solvers['StaticCoupledRBM']['structural_solver'] = 'RigidDynamicPrescribedStep'
SimInfo.solvers['StaticCoupledRBM']['structural_solver_settings'] = SimInfo.solvers['RigidDynamicPrescribedStep']
SimInfo.solvers['StaticCoupledRBM']['aero_solver'] = 'StaticUvlm'
SimInfo.solvers['StaticCoupledRBM']['aero_solver_settings'] = SimInfo.solvers['StaticUvlm']

SimInfo.solvers['StaticCoupledRBM']['tolerance'] = 1e-8
SimInfo.solvers['StaticCoupledRBM']['n_load_steps'] = 0
SimInfo.solvers['StaticCoupledRBM']['relaxation_factor'] = 0.

SimInfo.solvers['StaticUvlm']['num_cores'] = 8
SimInfo.solvers['StaticUvlm']['velocity_field_generator'] = 'SteadyVelocityField'
SimInfo.solvers['StaticUvlm']['velocity_field_input'] = SimInfo.solvers['SteadyVelocityField']

SimInfo.solvers['SaveData']['compress_float'] = True

# Only used for steady_simulation = False
SimInfo.solvers['StepUvlm']['convection_scheme'] = 3
SimInfo.solvers['StepUvlm']['num_cores'] = 8
SimInfo.solvers['StepUvlm']['velocity_field_generator'] = 'SteadyVelocityField'
SimInfo.solvers['StepUvlm']['velocity_field_input'] = SimInfo.solvers['SteadyVelocityField']

SimInfo.solvers['DynamicCoupled']['structural_solver'] = 'RigidDynamicPrescribedStep'
SimInfo.solvers['DynamicCoupled']['structural_solver_settings'] = SimInfo.solvers['RigidDynamicPrescribedStep']
SimInfo.solvers['DynamicCoupled']['aero_solver'] = 'StepUvlm'
SimInfo.solvers['DynamicCoupled']['aero_solver_settings'] = SimInfo.solvers['StepUvlm']
SimInfo.solvers['DynamicCoupled']['postprocessors'] = ['BeamPlot', 'AerogridPlot', 'Cleanup', 'SaveData']
SimInfo.solvers['DynamicCoupled']['postprocessors_settings'] = {'BeamPlot': SimInfo.solvers['BeamPlot'],
                                                             'AerogridPlot': SimInfo.solvers['AerogridPlot'],
                                                             'Cleanup': SimInfo.solvers['Cleanup'],
                                                             'SaveData': SimInfo.solvers['SaveData']}
SimInfo.solvers['DynamicCoupled']['minimum_steps'] = 0
SimInfo.solvers['DynamicCoupled']['include_unsteady_force_contribution'] = True
SimInfo.solvers['DynamicCoupled']['relaxation_factor'] = 0.
SimInfo.solvers['DynamicCoupled']['final_relaxation_factor'] = 0.
SimInfo.solvers['DynamicCoupled']['dynamic_relaxation'] = False
SimInfo.solvers['DynamicCoupled']['relaxation_steps'] = 0

# Define dynamic simulation (used regardless the value of "steady_simulation" variable)
SimInfo.define_num_steps(time_steps)
SimInfo.with_forced_vel = True
SimInfo.for_vel = np.zeros((time_steps,6), dtype=float)
SimInfo.for_vel[:,5] = rotation_velocity
SimInfo.for_acc = np.zeros((time_steps,6), dtype=float)
SimInfo.with_dynamic_forces = True
SimInfo.dynamic_forces = np.zeros((time_steps,rotor.StructuralInformation.num_node,6), dtype=float)

Generate simulation files

[9]:

gc.clean_test_files(SimInfo.solvers['SHARPy']['route'], SimInfo.solvers['SHARPy']['case'])
rotor.generate_h5_files(SimInfo.solvers['SHARPy']['route'], SimInfo.solvers['SHARPy']['case'])
SimInfo.generate_solver_file()
SimInfo.generate_dyn_file(time_steps)

Run SHARPy case¶

[10]:

sharpy_output = sharpy.sharpy_main.main(['', SimInfo.solvers['SHARPy']['route'] + SimInfo.solvers['SHARPy']['case'] + '.sharpy'])

--------------------------------------------------------------------------------
            ######  ##     ##    ###    ########  ########  ##    ##
           ##    ## ##     ##   ## ##   ##     ## ##     ##  ##  ##
           ##       ##     ##  ##   ##  ##     ## ##     ##   ####
            ######  ######### ##     ## ########  ########     ##
                 ## ##     ## ######### ##   ##   ##           ##
           ##    ## ##     ## ##     ## ##    ##  ##           ##
            ######  ##     ## ##     ## ##     ## ##           ##
--------------------------------------------------------------------------------
Aeroelastics Lab, Aeronautics Department.
    Copyright (c), Imperial College London.
    All rights reserved.
    License available at https://github.com/imperialcollegelondon/sharpy
Running SHARPy from /home/am3717/code/sharpy/docs/source/content/example_notebooks
SHARPy being run is in /home/am3717/code/sharpy
The branch being run is dev_setting_error
The version and commit hash are: v1.2.1-357-gbb7c4b4-bb7c4b4
SHARPy output folder set
     ./output//rotor/
Generating an instance of BeamLoader
Generating an instance of AerogridLoader
Variable shear_direction has no assigned value in the settings file.
    will default to the value: [1. 0. 0.]
Variable shear_exp has no assigned value in the settings file.
    will default to the value: 0.0
Variable h_ref has no assigned value in the settings file.
    will default to the value: 1.0
Variable h_corr has no assigned value in the settings file.
    will default to the value: 1.0
The aerodynamic grid contains 3 surfaces
  Surface 0, M=8, N=26
     Wake 0, M=450, N=26
  Surface 1, M=8, N=26
     Wake 1, M=450, N=26
  Surface 2, M=8, N=26
     Wake 2, M=450, N=26
  In total: 624 bound panels
  In total: 35100 wake panels
  Total number of panels = 35724
Generating an instance of StaticCoupledRBM
Generating an instance of RigidDynamicPrescribedStep
Generating an instance of StaticUvlm
i_step: 0, i_iter: 0
Resultant forces and moments: (array([2.54814836e-08, 5.82946313e-09, 2.09459948e+04]), array([1.53411975e-07, 3.24440634e-08, 5.96558361e+06]))
Generating an instance of DynamicCoupled
Generating an instance of RigidDynamicPrescribedStep
Generating an instance of StepUvlm
Generating an instance of BeamPlot
Generating an instance of AerogridPlot
Generating an instance of Cleanup
Generating an instance of SaveData



|=======|========|======|==============|==============|==============|==============|==============|
|  ts   |   t    | iter | struc ratio  |  iter time   | residual vel |  FoR_vel(x)  |  FoR_vel(z)  |
|=======|========|======|==============|==============|==============|==============|==============|
|   1   | 0.0551 |  1   |   0.000251   |  56.804234   |   0.000000   | 0.000000e+00 | 0.000000e+00 |
|   2   | 0.1102 |  1   |   0.000177   |  60.232129   |   0.000000   | 0.000000e+00 | 0.000000e+00 |
|   3   | 0.1653 |  1   |   0.000234   |  59.442611   |   0.000000   | 0.000000e+00 | 0.000000e+00 |
|   4   | 0.2204 |  1   |   0.000290   |  64.383652   |   0.000000   | 0.000000e+00 | 0.000000e+00 |
|   5   | 0.2755 |  1   |   0.000275   |  52.774436   |   0.000000   | 0.000000e+00 | 0.000000e+00 |
|   6   | 0.3306 |  1   |   0.000293   |  48.828457   |   0.000000   | 0.000000e+00 | 0.000000e+00 |
|   7   | 0.3857 |  1   |   0.000316   |  49.762675   |   0.000000   | 0.000000e+00 | 0.000000e+00 |
|   8   | 0.4408 |  1   |   0.000227   |  49.627842   |   0.000000   | 0.000000e+00 | 0.000000e+00 |
|   9   | 0.4959 |  1   |   0.000193   |  58.254770   |   0.000000   | 0.000000e+00 | 0.000000e+00 |
|  10   | 0.5510 |  1   |   0.000233   |  49.347327   |   0.000000   | 0.000000e+00 | 0.000000e+00 |
|  11   | 0.6061 |  1   |   0.000222   |  51.950339   |   0.000000   | 0.000000e+00 | 0.000000e+00 |
|  12   | 0.6612 |  1   |   0.000195   |  58.489715   |   0.000000   | 0.000000e+00 | 0.000000e+00 |
|  13   | 0.7163 |  1   |   0.000190   |  55.497720   |   0.000000   | 0.000000e+00 | 0.000000e+00 |
|  14   | 0.7713 |  1   |   0.000287   |  49.864559   |   0.000000   | 0.000000e+00 | 0.000000e+00 |
|  15   | 0.8264 |  1   |   0.000198   |  56.915311   |   0.000000   | 0.000000e+00 | 0.000000e+00 |
|  16   | 0.8815 |  1   |   0.000273   |  52.475129   |   0.000000   | 0.000000e+00 | 0.000000e+00 |
|  17   | 0.9366 |  1   |   0.000212   |  53.472742   |   0.000000   | 0.000000e+00 | 0.000000e+00 |
|  18   | 0.9917 |  1   |   0.000220   |  51.679432   |   0.000000   | 0.000000e+00 | 0.000000e+00 |
|  19   | 1.0468 |  1   |   0.000230   |  49.468328   |   0.000000   | 0.000000e+00 | 0.000000e+00 |
|  20   | 1.1019 |  1   |   0.000233   |  48.806374   |   0.000000   | 0.000000e+00 | 0.000000e+00 |
|  21   | 1.1570 |  1   |   0.000214   |  51.507987   |   0.000000   | 0.000000e+00 | 0.000000e+00 |
|  22   | 1.2121 |  1   |   0.000152   |  75.114590   |   0.000000   | 0.000000e+00 | 0.000000e+00 |
|  23   | 1.2672 |  1   |   0.000178   |  63.764692   |   0.000000   | 0.000000e+00 | 0.000000e+00 |
|  24   | 1.3223 |  1   |   0.000232   |  48.927093   |   0.000000   | 0.000000e+00 | 0.000000e+00 |
|  25   | 1.3774 |  1   |   0.000205   |  55.065928   |   0.000000   | 0.000000e+00 | 0.000000e+00 |
|  26   | 1.4325 |  1   |   0.000209   |  53.747631   |   0.000000   | 0.000000e+00 | 0.000000e+00 |
|  27   | 1.4876 |  1   |   0.000215   |  65.487096   |   0.000000   | 0.000000e+00 | 0.000000e+00 |
|  28   | 1.5427 |  1   |   0.000366   |  52.159906   |   0.000000   | 0.000000e+00 | 0.000000e+00 |
|  29   | 1.5978 |  1   |   0.000286   |  50.405248   |   0.000000   | 0.000000e+00 | 0.000000e+00 |
|  30   | 1.6529 |  1   |   0.000243   |  53.903698   |   0.000000   | 0.000000e+00 | 0.000000e+00 |
|  31   | 1.7080 |  1   |   0.000189   |  55.762230   |   0.000000   | 0.000000e+00 | 0.000000e+00 |
|  32   | 1.7631 |  1   |   0.000230   |  49.679477   |   0.000000   | 0.000000e+00 | 0.000000e+00 |
|  33   | 1.8182 |  1   |   0.000267   |  49.140923   |   0.000000   | 0.000000e+00 | 0.000000e+00 |
|  34   | 1.8733 |  1   |   0.000217   |  51.686471   |   0.000000   | 0.000000e+00 | 0.000000e+00 |
|  35   | 1.9284 |  1   |   0.000227   |  49.924635   |   0.000000   | 0.000000e+00 | 0.000000e+00 |
|  36   | 1.9835 |  1   |   0.000290   |  49.295199   |   0.000000   | 0.000000e+00 | 0.000000e+00 |
|  37   | 2.0386 |  1   |   0.000231   |  49.096186   |   0.000000   | 0.000000e+00 | 0.000000e+00 |
|  38   | 2.0937 |  1   |   0.000267   |  53.824351   |   0.000000   | 0.000000e+00 | 0.000000e+00 |
|  39   | 2.1488 |  1   |   0.000274   |  50.208116   |   0.000000   | 0.000000e+00 | 0.000000e+00 |
|  40   | 2.2039 |  1   |   0.000164   |  68.475332   |   0.000000   | 0.000000e+00 | 0.000000e+00 |
|  41   | 2.2590 |  1   |   0.000219   |  51.476537   |   0.000000   | 0.000000e+00 | 0.000000e+00 |
|  42   | 2.3140 |  1   |   0.000258   |  50.780835   |   0.000000   | 0.000000e+00 | 0.000000e+00 |
|  43   | 2.3691 |  1   |   0.000205   |  54.508414   |   0.000000   | 0.000000e+00 | 0.000000e+00 |
|  44   | 2.4242 |  1   |   0.000229   |  49.643120   |   0.000000   | 0.000000e+00 | 0.000000e+00 |
|  45   | 2.4793 |  1   |   0.000423   |  49.510440   |   0.000000   | 0.000000e+00 | 0.000000e+00 |
|  46   | 2.5344 |  1   |   0.000229   |  49.386177   |   0.000000   | 0.000000e+00 | 0.000000e+00 |
|  47   | 2.5895 |  1   |   0.000224   |  51.096846   |   0.000000   | 0.000000e+00 | 0.000000e+00 |
|  48   | 2.6446 |  1   |   0.000429   |  67.054611   |   0.000000   | 0.000000e+00 | 0.000000e+00 |
|  49   | 2.6997 |  1   |   0.000674   |  49.034094   |   0.000000   | 0.000000e+00 | 0.000000e+00 |
|  50   | 2.7548 |  1   |   0.000473   |  67.770922   |   0.000000   | 0.000000e+00 | 0.000000e+00 |
|  51   | 2.8099 |  1   |   0.000623   |  52.057162   |   0.000000   | 0.000000e+00 | 0.000000e+00 |
|  52   | 2.8650 |  1   |   0.000197   |  57.338530   |   0.000000   | 0.000000e+00 | 0.000000e+00 |
|  53   | 2.9201 |  1   |   0.000211   |  53.323214   |   0.000000   | 0.000000e+00 | 0.000000e+00 |
|  54   | 2.9752 |  1   |   0.000228   |  49.446313   |   0.000000   | 0.000000e+00 | 0.000000e+00 |
|  55   | 3.0303 |  1   |   0.000226   |  50.531894   |   0.000000   | 0.000000e+00 | 0.000000e+00 |
|  56   | 3.0854 |  1   |   0.000216   |  49.277498   |   0.000000   | 0.000000e+00 | 0.000000e+00 |
|  57   | 3.1405 |  1   |   0.000229   |  49.378688   |   0.000000   | 0.000000e+00 | 0.000000e+00 |
|  58   | 3.1956 |  1   |   0.000456   |  49.096131   |   0.000000   | 0.000000e+00 | 0.000000e+00 |
|  59   | 3.2507 |  1   |   0.000638   |  49.407950   |   0.000000   | 0.000000e+00 | 0.000000e+00 |
|  60   | 3.3058 |  1   |   0.000644   |  49.355487   |   0.000000   | 0.000000e+00 | 0.000000e+00 |
|  61   | 3.3609 |  1   |   0.000429   |  52.381709   |   0.000000   | 0.000000e+00 | 0.000000e+00 |
|  62   | 3.4160 |  1   |   0.000227   |  50.244867   |   0.000000   | 0.000000e+00 | 0.000000e+00 |
|  63   | 3.4711 |  1   |   0.000226   |  50.090238   |   0.000000   | 0.000000e+00 | 0.000000e+00 |
|  64   | 3.5262 |  1   |   0.000221   |  50.890983   |   0.000000   | 0.000000e+00 | 0.000000e+00 |
|  65   | 3.5813 |  1   |   0.000292   |  52.582257   |   0.000000   | 0.000000e+00 | 0.000000e+00 |
|  66   | 3.6364 |  1   |   0.000229   |  48.970770   |   0.000000   | 0.000000e+00 | 0.000000e+00 |
|  67   | 3.6915 |  1   |   0.000211   |  53.921150   |   0.000000   | 0.000000e+00 | 0.000000e+00 |
|  68   | 3.7466 |  1   |   0.000276   |  49.332785   |   0.000000   | 0.000000e+00 | 0.000000e+00 |
|  69   | 3.8017 |  1   |   0.000405   |  49.861912   |   0.000000   | 0.000000e+00 | 0.000000e+00 |
|  70   | 3.8567 |  1   |   0.000678   |  50.109032   |   0.000000   | 0.000000e+00 | 0.000000e+00 |
|  71   | 3.9118 |  1   |   0.000627   |  50.052867   |   0.000000   | 0.000000e+00 | 0.000000e+00 |
|  72   | 3.9669 |  1   |   0.000231   |  49.125818   |   0.000000   | 0.000000e+00 | 0.000000e+00 |
|  73   | 4.0220 |  1   |   0.000215   |  52.475354   |   0.000000   | 0.000000e+00 | 0.000000e+00 |
|  74   | 4.0771 |  1   |   0.000302   |  51.044905   |   0.000000   | 0.000000e+00 | 0.000000e+00 |
|  75   | 4.1322 |  1   |   0.000232   |  49.154350   |   0.000000   | 0.000000e+00 | 0.000000e+00 |
|  76   | 4.1873 |  1   |   0.000263   |  49.587372   |   0.000000   | 0.000000e+00 | 0.000000e+00 |
|  77   | 4.2424 |  1   |   0.000382   |  48.959923   |   0.000000   | 0.000000e+00 | 0.000000e+00 |
|  78   | 4.2975 |  1   |   0.000207   |  54.498279   |   0.000000   | 0.000000e+00 | 0.000000e+00 |
|  79   | 4.3526 |  1   |   0.000209   |  55.866552   |   0.000000   | 0.000000e+00 | 0.000000e+00 |
|  80   | 4.4077 |  1   |   0.000222   |  50.659905   |   0.000000   | 0.000000e+00 | 0.000000e+00 |
|  81   | 4.4628 |  1   |   0.000232   |  49.229321   |   0.000000   | 0.000000e+00 | 0.000000e+00 |
|  82   | 4.5179 |  1   |   0.000230   |  49.197802   |   0.000000   | 0.000000e+00 | 0.000000e+00 |
|  83   | 4.5730 |  1   |   0.000244   |  58.532026   |   0.000000   | 0.000000e+00 | 0.000000e+00 |
|  84   | 4.6281 |  1   |   0.000229   |  49.010943   |   0.000000   | 0.000000e+00 | 0.000000e+00 |
|  85   | 4.6832 |  1   |   0.000236   |  49.181471   |   0.000000   | 0.000000e+00 | 0.000000e+00 |
|  86   | 4.7383 |  1   |   0.000234   |  50.404876   |   0.000000   | 0.000000e+00 | 0.000000e+00 |
|  87   | 4.7934 |  1   |   0.000223   |  50.863604   |   0.000000   | 0.000000e+00 | 0.000000e+00 |
|  88   | 4.8485 |  1   |   0.000232   |  49.386036   |   0.000000   | 0.000000e+00 | 0.000000e+00 |
|  89   | 4.9036 |  1   |   0.000292   |  48.966426   |   0.000000   | 0.000000e+00 | 0.000000e+00 |
|  90   | 4.9587 |  1   |   0.000204   |  55.176201   |   0.000000   | 0.000000e+00 | 0.000000e+00 |
|  91   | 5.0138 |  1   |   0.000381   |  49.363449   |   0.000000   | 0.000000e+00 | 0.000000e+00 |
|  92   | 5.0689 |  1   |   0.000254   |  51.436284   |   0.000000   | 0.000000e+00 | 0.000000e+00 |
|  93   | 5.1240 |  1   |   0.000427   |  54.203413   |   0.000000   | 0.000000e+00 | 0.000000e+00 |
|  94   | 5.1791 |  1   |   0.000278   |  51.088371   |   0.000000   | 0.000000e+00 | 0.000000e+00 |
|  95   | 5.2342 |  1   |   0.000221   |  51.267494   |   0.000000   | 0.000000e+00 | 0.000000e+00 |
|  96   | 5.2893 |  1   |   0.000220   |  51.246126   |   0.000000   | 0.000000e+00 | 0.000000e+00 |
|  97   | 5.3444 |  1   |   0.000261   |  50.044367   |   0.000000   | 0.000000e+00 | 0.000000e+00 |
|  98   | 5.3994 |  1   |   0.000237   |  48.928531   |   0.000000   | 0.000000e+00 | 0.000000e+00 |
|  99   | 5.4545 |  1   |   0.000230   |  48.869230   |   0.000000   | 0.000000e+00 | 0.000000e+00 |
|  100  | 5.5096 |  1   |   0.000229   |  49.124024   |   0.000000   | 0.000000e+00 | 0.000000e+00 |
|  101  | 5.5647 |  1   |   0.000230   |  49.170615   |   0.000000   | 0.000000e+00 | 0.000000e+00 |
|  102  | 5.6198 |  1   |   0.000227   |  51.072559   |   0.000000   | 0.000000e+00 | 0.000000e+00 |
|  103  | 5.6749 |  1   |   0.000230   |  49.106752   |   0.000000   | 0.000000e+00 | 0.000000e+00 |
|  104  | 5.7300 |  1   |   0.000230   |  49.157374   |   0.000000   | 0.000000e+00 | 0.000000e+00 |
|  105  | 5.7851 |  1   |   0.000230   |  48.729726   |   0.000000   | 0.000000e+00 | 0.000000e+00 |
|  106  | 5.8402 |  1   |   0.000230   |  49.307981   |   0.000000   | 0.000000e+00 | 0.000000e+00 |
|  107  | 5.8953 |  1   |   0.000232   |  49.303297   |   0.000000   | 0.000000e+00 | 0.000000e+00 |
|  108  | 5.9504 |  1   |   0.000228   |  49.388923   |   0.000000   | 0.000000e+00 | 0.000000e+00 |
|  109  | 6.0055 |  1   |   0.000251   |  49.505395   |   0.000000   | 0.000000e+00 | 0.000000e+00 |
|  110  | 6.0606 |  1   |   0.000294   |  49.671154   |   0.000000   | 0.000000e+00 | 0.000000e+00 |
|  111  | 6.1157 |  1   |   0.000637   |  49.106910   |   0.000000   | 0.000000e+00 | 0.000000e+00 |
|  112  | 6.1708 |  1   |   0.000668   |  49.600747   |   0.000000   | 0.000000e+00 | 0.000000e+00 |
|  113  | 6.2259 |  1   |   0.000694   |  48.788273   |   0.000000   | 0.000000e+00 | 0.000000e+00 |
|  114  | 6.2810 |  1   |   0.000605   |  49.259812   |   0.000000   | 0.000000e+00 | 0.000000e+00 |
|  115  | 6.3361 |  1   |   0.000235   |  49.096078   |   0.000000   | 0.000000e+00 | 0.000000e+00 |
|  116  | 6.3912 |  1   |   0.000235   |  48.627974   |   0.000000   | 0.000000e+00 | 0.000000e+00 |
|  117  | 6.4463 |  1   |   0.000231   |  48.687230   |   0.000000   | 0.000000e+00 | 0.000000e+00 |
|  118  | 6.5014 |  1   |   0.000261   |  50.182136   |   0.000000   | 0.000000e+00 | 0.000000e+00 |
|  119  | 6.5565 |  1   |   0.000220   |  51.480792   |   0.000000   | 0.000000e+00 | 0.000000e+00 |
|  120  | 6.6116 |  1   |   0.000212   |  52.759361   |   0.000000   | 0.000000e+00 | 0.000000e+00 |
|  121  | 6.6667 |  1   |   0.000234   |  61.395446   |   0.000000   | 0.000000e+00 | 0.000000e+00 |
|  122  | 6.7218 |  1   |   0.000196   |  57.312678   |   0.000000   | 0.000000e+00 | 0.000000e+00 |
|  123  | 6.7769 |  1   |   0.000196   |  57.288595   |   0.000000   | 0.000000e+00 | 0.000000e+00 |
|  124  | 6.8320 |  1   |   0.000219   |  52.597533   |   0.000000   | 0.000000e+00 | 0.000000e+00 |
|  125  | 6.8871 |  1   |   0.000279   |  51.215139   |   0.000000   | 0.000000e+00 | 0.000000e+00 |
|  126  | 6.9421 |  1   |   0.000232   |  48.835388   |   0.000000   | 0.000000e+00 | 0.000000e+00 |
|  127  | 6.9972 |  1   |   0.000228   |  49.442145   |   0.000000   | 0.000000e+00 | 0.000000e+00 |
|  128  | 7.0523 |  1   |   0.000232   |  49.943434   |   0.000000   | 0.000000e+00 | 0.000000e+00 |
|  129  | 7.1074 |  1   |   0.000265   |  48.943739   |   0.000000   | 0.000000e+00 | 0.000000e+00 |
|  130  | 7.1625 |  1   |   0.000233   |  48.552538   |   0.000000   | 0.000000e+00 | 0.000000e+00 |
|  131  | 7.2176 |  1   |   0.000235   |  48.576140   |   0.000000   | 0.000000e+00 | 0.000000e+00 |
|  132  | 7.2727 |  1   |   0.000227   |  50.729522   |   0.000000   | 0.000000e+00 | 0.000000e+00 |
|  133  | 7.3278 |  1   |   0.000199   |  57.042671   |   0.000000   | 0.000000e+00 | 0.000000e+00 |
|  134  | 7.3829 |  1   |   0.000229   |  49.647710   |   0.000000   | 0.000000e+00 | 0.000000e+00 |
|  135  | 7.4380 |  1   |   0.000377   |  53.191903   |   0.000000   | 0.000000e+00 | 0.000000e+00 |
|  136  | 7.4931 |  1   |   0.000666   |  50.832518   |   0.000000   | 0.000000e+00 | 0.000000e+00 |
|  137  | 7.5482 |  1   |   0.000560   |  52.129724   |   0.000000   | 0.000000e+00 | 0.000000e+00 |
|  138  | 7.6033 |  1   |   0.000593   |  48.825715   |   0.000000   | 0.000000e+00 | 0.000000e+00 |
|  139  | 7.6584 |  1   |   0.000231   |  49.220850   |   0.000000   | 0.000000e+00 | 0.000000e+00 |
|  140  | 7.7135 |  1   |   0.000347   |  64.642874   |   0.000000   | 0.000000e+00 | 0.000000e+00 |
|  141  | 7.7686 |  1   |   0.000615   |  54.791703   |   0.000000   | 0.000000e+00 | 0.000000e+00 |
|  142  | 7.8237 |  1   |   0.000557   |  55.332506   |   0.000000   | 0.000000e+00 | 0.000000e+00 |
|  143  | 7.8788 |  1   |   0.000649   |  51.958248   |   0.000000   | 0.000000e+00 | 0.000000e+00 |
|  144  | 7.9339 |  1   |   0.000431   |  52.224418   |   0.000000   | 0.000000e+00 | 0.000000e+00 |
|  145  | 7.9890 |  1   |   0.000532   |  60.207598   |   0.000000   | 0.000000e+00 | 0.000000e+00 |
|  146  | 8.0441 |  1   |   0.000267   |  48.557877   |   0.000000   | 0.000000e+00 | 0.000000e+00 |
|  147  | 8.0992 |  1   |   0.000283   |  50.720948   |   0.000000   | 0.000000e+00 | 0.000000e+00 |
|  148  | 8.1543 |  1   |   0.000264   |  54.107664   |   0.000000   | 0.000000e+00 | 0.000000e+00 |
|  149  | 8.2094 |  1   |   0.000225   |  49.998135   |   0.000000   | 0.000000e+00 | 0.000000e+00 |
|  150  | 8.2645 |  1   |   0.000297   |  48.723537   |   0.000000   | 0.000000e+00 | 0.000000e+00 |
|  151  | 8.3196 |  1   |   0.000229   |  48.991915   |   0.000000   | 0.000000e+00 | 0.000000e+00 |
|  152  | 8.3747 |  1   |   0.000223   |  50.525580   |   0.000000   | 0.000000e+00 | 0.000000e+00 |
|  153  | 8.4298 |  1   |   0.000372   |  48.880427   |   0.000000   | 0.000000e+00 | 0.000000e+00 |
|  154  | 8.4848 |  1   |   0.000232   |  48.851891   |   0.000000   | 0.000000e+00 | 0.000000e+00 |
|  155  | 8.5399 |  1   |   0.000231   |  48.731096   |   0.000000   | 0.000000e+00 | 0.000000e+00 |
|  156  | 8.5950 |  1   |   0.000209   |  53.669843   |   0.000000   | 0.000000e+00 | 0.000000e+00 |
|  157  | 8.6501 |  1   |   0.000203   |  55.525421   |   0.000000   | 0.000000e+00 | 0.000000e+00 |
|  158  | 8.7052 |  1   |   0.000216   |  48.727216   |   0.000000   | 0.000000e+00 | 0.000000e+00 |
|  159  | 8.7603 |  1   |   0.000240   |  52.981239   |   0.000000   | 0.000000e+00 | 0.000000e+00 |
|  160  | 8.8154 |  1   |   0.000294   |  50.516618   |   0.000000   | 0.000000e+00 | 0.000000e+00 |
|  161  | 8.8705 |  1   |   0.000198   |  56.957355   |   0.000000   | 0.000000e+00 | 0.000000e+00 |
|  162  | 8.9256 |  1   |   0.000268   |  48.780843   |   0.000000   | 0.000000e+00 | 0.000000e+00 |
|  163  | 8.9807 |  1   |   0.000228   |  49.216988   |   0.000000   | 0.000000e+00 | 0.000000e+00 |
|  164  | 9.0358 |  1   |   0.000228   |  49.020854   |   0.000000   | 0.000000e+00 | 0.000000e+00 |
|  165  | 9.0909 |  1   |   0.000236   |  48.516192   |   0.000000   | 0.000000e+00 | 0.000000e+00 |
|  166  | 9.1460 |  1   |   0.000275   |  48.936249   |   0.000000   | 0.000000e+00 | 0.000000e+00 |
|  167  | 9.2011 |  1   |   0.000229   |  48.984558   |   0.000000   | 0.000000e+00 | 0.000000e+00 |
|  168  | 9.2562 |  1   |   0.000228   |  50.011495   |   0.000000   | 0.000000e+00 | 0.000000e+00 |
|  169  | 9.3113 |  1   |   0.000230   |  49.079509   |   0.000000   | 0.000000e+00 | 0.000000e+00 |
|  170  | 9.3664 |  1   |   0.000232   |  48.940832   |   0.000000   | 0.000000e+00 | 0.000000e+00 |
|  171  | 9.4215 |  1   |   0.000266   |  48.539668   |   0.000000   | 0.000000e+00 | 0.000000e+00 |
|  172  | 9.4766 |  1   |   0.000329   |  48.707876   |   0.000000   | 0.000000e+00 | 0.000000e+00 |
|  173  | 9.5317 |  1   |   0.000344   |  49.214713   |   0.000000   | 0.000000e+00 | 0.000000e+00 |
|  174  | 9.5868 |  1   |   0.000221   |  50.762513   |   0.000000   | 0.000000e+00 | 0.000000e+00 |
|  175  | 9.6419 |  1   |   0.000336   |  55.302099   |   0.000000   | 0.000000e+00 | 0.000000e+00 |
|  176  | 9.6970 |  1   |   0.000217   |  48.736059   |   0.000000   | 0.000000e+00 | 0.000000e+00 |
|  177  | 9.7521 |  1   |   0.000231   |  48.750098   |   0.000000   | 0.000000e+00 | 0.000000e+00 |
|  178  | 9.8072 |  1   |   0.000229   |  48.794680   |   0.000000   | 0.000000e+00 | 0.000000e+00 |
|  179  | 9.8623 |  1   |   0.000225   |  50.519301   |   0.000000   | 0.000000e+00 | 0.000000e+00 |
|  180  | 9.9174 |  1   |   0.000225   |  49.930789   |   0.000000   | 0.000000e+00 | 0.000000e+00 |
|  181  | 9.9725 |  1   |   0.000265   |  50.077232   |   0.000000   | 0.000000e+00 | 0.000000e+00 |
|  182  |10.0275 |  1   |   0.000285   |  50.327260   |   0.000000   | 0.000000e+00 | 0.000000e+00 |
|  183  |10.0826 |  1   |   0.000319   |  50.215326   |   0.000000   | 0.000000e+00 | 0.000000e+00 |
|  184  |10.1377 |  1   |   0.000288   |  48.350528   |   0.000000   | 0.000000e+00 | 0.000000e+00 |
|  185  |10.1928 |  1   |   0.000233   |  48.372314   |   0.000000   | 0.000000e+00 | 0.000000e+00 |
|  186  |10.2479 |  1   |   0.000232   |  49.104808   |   0.000000   | 0.000000e+00 | 0.000000e+00 |
|  187  |10.3030 |  1   |   0.000292   |  48.972235   |   0.000000   | 0.000000e+00 | 0.000000e+00 |
|  188  |10.3581 |  1   |   0.000655   |  48.972110   |   0.000000   | 0.000000e+00 | 0.000000e+00 |
|  189  |10.4132 |  1   |   0.000233   |  48.457135   |   0.000000   | 0.000000e+00 | 0.000000e+00 |
|  190  |10.4683 |  1   |   0.000229   |  49.120724   |   0.000000   | 0.000000e+00 | 0.000000e+00 |
|  191  |10.5234 |  1   |   0.000222   |  51.106254   |   0.000000   | 0.000000e+00 | 0.000000e+00 |
|  192  |10.5785 |  1   |   0.000232   |  48.794933   |   0.000000   | 0.000000e+00 | 0.000000e+00 |
|  193  |10.6336 |  1   |   0.000292   |  49.537670   |   0.000000   | 0.000000e+00 | 0.000000e+00 |
|  194  |10.6887 |  1   |   0.000230   |  49.236260   |   0.000000   | 0.000000e+00 | 0.000000e+00 |
|  195  |10.7438 |  1   |   0.000603   |  52.529460   |   0.000000   | 0.000000e+00 | 0.000000e+00 |
|  196  |10.7989 |  1   |   0.000697   |  48.383116   |   0.000000   | 0.000000e+00 | 0.000000e+00 |
|  197  |10.8540 |  1   |   0.000347   |  48.288813   |   0.000000   | 0.000000e+00 | 0.000000e+00 |
|  198  |10.9091 |  1   |   0.000227   |  49.550065   |   0.000000   | 0.000000e+00 | 0.000000e+00 |
|  199  |10.9642 |  1   |   0.000216   |  48.675859   |   0.000000   | 0.000000e+00 | 0.000000e+00 |
|  200  |11.0193 |  1   |   0.000231   |  49.152487   |   0.000000   | 0.000000e+00 | 0.000000e+00 |
|  201  |11.0744 |  1   |   0.000235   |  48.778197   |   0.000000   | 0.000000e+00 | 0.000000e+00 |
|  202  |11.1295 |  1   |   0.000373   |  57.791454   |   0.000000   | 0.000000e+00 | 0.000000e+00 |
|  203  |11.1846 |  1   |   0.000582   |  57.753718   |   0.000000   | 0.000000e+00 | 0.000000e+00 |
|  204  |11.2397 |  1   |   0.000451   |  48.675944   |   0.000000   | 0.000000e+00 | 0.000000e+00 |
|  205  |11.2948 |  1   |   0.000235   |  48.274112   |   0.000000   | 0.000000e+00 | 0.000000e+00 |
|  206  |11.3499 |  1   |   0.000232   |  48.917871   |   0.000000   | 0.000000e+00 | 0.000000e+00 |
|  207  |11.4050 |  1   |   0.000293   |  49.482251   |   0.000000   | 0.000000e+00 | 0.000000e+00 |
|  208  |11.4601 |  1   |   0.000297   |  48.304309   |   0.000000   | 0.000000e+00 | 0.000000e+00 |
|  209  |11.5152 |  1   |   0.000263   |  49.162448   |   0.000000   | 0.000000e+00 | 0.000000e+00 |
|  210  |11.5702 |  1   |   0.000227   |  50.845835   |   0.000000   | 0.000000e+00 | 0.000000e+00 |
|  211  |11.6253 |  1   |   0.000232   |  48.809305   |   0.000000   | 0.000000e+00 | 0.000000e+00 |
|  212  |11.6804 |  1   |   0.000293   |  48.950330   |   0.000000   | 0.000000e+00 | 0.000000e+00 |
|  213  |11.7355 |  1   |   0.000298   |  48.481146   |   0.000000   | 0.000000e+00 | 0.000000e+00 |
|  214  |11.7906 |  1   |   0.000234   |  48.222179   |   0.000000   | 0.000000e+00 | 0.000000e+00 |
|  215  |11.8457 |  1   |   0.000225   |  48.228569   |   0.000000   | 0.000000e+00 | 0.000000e+00 |
|  216  |11.9008 |  1   |   0.000218   |  48.173668   |   0.000000   | 0.000000e+00 | 0.000000e+00 |
|  217  |11.9559 |  1   |   0.000230   |  48.496389   |   0.000000   | 0.000000e+00 | 0.000000e+00 |
|  218  |12.0110 |  1   |   0.000212   |  53.240908   |   0.000000   | 0.000000e+00 | 0.000000e+00 |
|  219  |12.0661 |  1   |   0.000224   |  51.372105   |   0.000000   | 0.000000e+00 | 0.000000e+00 |
|  220  |12.1212 |  1   |   0.000419   |  49.490550   |   0.000000   | 0.000000e+00 | 0.000000e+00 |
|  221  |12.1763 |  1   |   0.000459   |  49.025501   |   0.000000   | 0.000000e+00 | 0.000000e+00 |
|  222  |12.2314 |  1   |   0.000210   |  53.157816   |   0.000000   | 0.000000e+00 | 0.000000e+00 |
|  223  |12.2865 |  1   |   0.000233   |  48.099241   |   0.000000   | 0.000000e+00 | 0.000000e+00 |
|  224  |12.3416 |  1   |   0.000241   |  59.450060   |   0.000000   | 0.000000e+00 | 0.000000e+00 |
|  225  |12.3967 |  1   |   0.000224   |  50.153783   |   0.000000   | 0.000000e+00 | 0.000000e+00 |
|  226  |12.4518 |  1   |   0.000169   |  67.550282   |   0.000000   | 0.000000e+00 | 0.000000e+00 |
|  227  |12.5069 |  1   |   0.000635   |  51.062920   |   0.000000   | 0.000000e+00 | 0.000000e+00 |
|  228  |12.5620 |  1   |   0.000233   |  48.498621   |   0.000000   | 0.000000e+00 | 0.000000e+00 |
|  229  |12.6171 |  1   |   0.000647   |  48.837410   |   0.000000   | 0.000000e+00 | 0.000000e+00 |
|  230  |12.6722 |  1   |   0.000685   |  48.524438   |   0.000000   | 0.000000e+00 | 0.000000e+00 |
|  231  |12.7273 |  1   |   0.000645   |  47.972726   |   0.000000   | 0.000000e+00 | 0.000000e+00 |
|  232  |12.7824 |  1   |   0.000698   |  48.332650   |   0.000000   | 0.000000e+00 | 0.000000e+00 |
|  233  |12.8375 |  1   |   0.000291   |  49.344210   |   0.000000   | 0.000000e+00 | 0.000000e+00 |
|  234  |12.8926 |  1   |   0.000231   |  48.621207   |   0.000000   | 0.000000e+00 | 0.000000e+00 |
|  235  |12.9477 |  1   |   0.000211   |  53.099352   |   0.000000   | 0.000000e+00 | 0.000000e+00 |
|  236  |13.0028 |  1   |   0.000231   |  48.779174   |   0.000000   | 0.000000e+00 | 0.000000e+00 |
|  237  |13.0579 |  1   |   0.000221   |  52.508748   |   0.000000   | 0.000000e+00 | 0.000000e+00 |
|  238  |13.1129 |  1   |   0.000197   |  57.071891   |   0.000000   | 0.000000e+00 | 0.000000e+00 |
|  239  |13.1680 |  1   |   0.000232   |  48.635785   |   0.000000   | 0.000000e+00 | 0.000000e+00 |
|  240  |13.2231 |  1   |   0.000228   |  49.653808   |   0.000000   | 0.000000e+00 | 0.000000e+00 |
|  241  |13.2782 |  1   |   0.000232   |  49.880283   |   0.000000   | 0.000000e+00 | 0.000000e+00 |
|  242  |13.3333 |  1   |   0.000219   |  50.989864   |   0.000000   | 0.000000e+00 | 0.000000e+00 |
|  243  |13.3884 |  1   |   0.000218   |  51.562733   |   0.000000   | 0.000000e+00 | 0.000000e+00 |
|  244  |13.4435 |  1   |   0.000175   |  60.525012   |   0.000000   | 0.000000e+00 | 0.000000e+00 |
|  245  |13.4986 |  1   |   0.000208   |  53.714735   |   0.000000   | 0.000000e+00 | 0.000000e+00 |
|  246  |13.5537 |  1   |   0.000188   |  60.430187   |   0.000000   | 0.000000e+00 | 0.000000e+00 |
|  247  |13.6088 |  1   |   0.000580   |  55.369422   |   0.000000   | 0.000000e+00 | 0.000000e+00 |
|  248  |13.6639 |  1   |   0.000376   |  52.945929   |   0.000000   | 0.000000e+00 | 0.000000e+00 |
|  249  |13.7190 |  1   |   0.000221   |  50.621281   |   0.000000   | 0.000000e+00 | 0.000000e+00 |
|  250  |13.7741 |  1   |   0.000222   |  51.930545   |   0.000000   | 0.000000e+00 | 0.000000e+00 |
|  251  |13.8292 |  1   |   0.000203   |  55.750479   |   0.000000   | 0.000000e+00 | 0.000000e+00 |
|  252  |13.8843 |  1   |   0.000228   |  49.736165   |   0.000000   | 0.000000e+00 | 0.000000e+00 |
|  253  |13.9394 |  1   |   0.000239   |  48.733683   |   0.000000   | 0.000000e+00 | 0.000000e+00 |
|  254  |13.9945 |  1   |   0.000228   |  48.860423   |   0.000000   | 0.000000e+00 | 0.000000e+00 |
|  255  |14.0496 |  1   |   0.000234   |  48.158656   |   0.000000   | 0.000000e+00 | 0.000000e+00 |
|  256  |14.1047 |  1   |   0.000222   |  48.995898   |   0.000000   | 0.000000e+00 | 0.000000e+00 |
|  257  |14.1598 |  1   |   0.000181   |  61.699569   |   0.000000   | 0.000000e+00 | 0.000000e+00 |
|  258  |14.2149 |  1   |   0.000217   |  51.853006   |   0.000000   | 0.000000e+00 | 0.000000e+00 |
|  259  |14.2700 |  1   |   0.000231   |  48.370357   |   0.000000   | 0.000000e+00 | 0.000000e+00 |
|  260  |14.3251 |  1   |   0.000231   |  48.536412   |   0.000000   | 0.000000e+00 | 0.000000e+00 |
|  261  |14.3802 |  1   |   0.000231   |  48.706123   |   0.000000   | 0.000000e+00 | 0.000000e+00 |
|  262  |14.4353 |  1   |   0.000321   |  48.889459   |   0.000000   | 0.000000e+00 | 0.000000e+00 |
|  263  |14.4904 |  1   |   0.000232   |  48.413389   |   0.000000   | 0.000000e+00 | 0.000000e+00 |
|  264  |14.5455 |  1   |   0.000183   |  61.637297   |   0.000000   | 0.000000e+00 | 0.000000e+00 |
|  265  |14.6006 |  1   |   0.000266   |  51.996729   |   0.000000   | 0.000000e+00 | 0.000000e+00 |
|  266  |14.6556 |  1   |   0.000440   |  51.124606   |   0.000000   | 0.000000e+00 | 0.000000e+00 |
|  267  |14.7107 |  1   |   0.000206   |  54.482224   |   0.000000   | 0.000000e+00 | 0.000000e+00 |
|  268  |14.7658 |  1   |   0.000227   |  49.396567   |   0.000000   | 0.000000e+00 | 0.000000e+00 |
|  269  |14.8209 |  1   |   0.000216   |  48.399076   |   0.000000   | 0.000000e+00 | 0.000000e+00 |
|  270  |14.8760 |  1   |   0.000225   |  49.932379   |   0.000000   | 0.000000e+00 | 0.000000e+00 |
|  271  |14.9311 |  1   |   0.000229   |  48.837313   |   0.000000   | 0.000000e+00 | 0.000000e+00 |
|  272  |14.9862 |  1   |   0.000234   |  48.303012   |   0.000000   | 0.000000e+00 | 0.000000e+00 |
|  273  |15.0413 |  1   |   0.000297   |  48.557886   |   0.000000   | 0.000000e+00 | 0.000000e+00 |
|  274  |15.0964 |  1   |   0.000232   |  48.462573   |   0.000000   | 0.000000e+00 | 0.000000e+00 |
|  275  |15.1515 |  1   |   0.000235   |  48.074680   |   0.000000   | 0.000000e+00 | 0.000000e+00 |
|  276  |15.2066 |  1   |   0.000237   |  47.978437   |   0.000000   | 0.000000e+00 | 0.000000e+00 |
|  277  |15.2617 |  1   |   0.000218   |  48.799704   |   0.000000   | 0.000000e+00 | 0.000000e+00 |
|  278  |15.3168 |  1   |   0.000180   |  61.907709   |   0.000000   | 0.000000e+00 | 0.000000e+00 |
|  279  |15.3719 |  1   |   0.000223   |  50.256748   |   0.000000   | 0.000000e+00 | 0.000000e+00 |
|  280  |15.4270 |  1   |   0.000229   |  49.243923   |   0.000000   | 0.000000e+00 | 0.000000e+00 |
|  281  |15.4821 |  1   |   0.000453   |  49.924487   |   0.000000   | 0.000000e+00 | 0.000000e+00 |
|  282  |15.5372 |  1   |   0.000350   |  49.501563   |   0.000000   | 0.000000e+00 | 0.000000e+00 |
|  283  |15.5923 |  1   |   0.000234   |  48.333380   |   0.000000   | 0.000000e+00 | 0.000000e+00 |
|  284  |15.6474 |  1   |   0.000223   |  50.163273   |   0.000000   | 0.000000e+00 | 0.000000e+00 |
|  285  |15.7025 |  1   |   0.000225   |  50.068699   |   0.000000   | 0.000000e+00 | 0.000000e+00 |
|  286  |15.7576 |  1   |   0.000229   |  48.458516   |   0.000000   | 0.000000e+00 | 0.000000e+00 |
|  287  |15.8127 |  1   |   0.000231   |  49.273332   |   0.000000   | 0.000000e+00 | 0.000000e+00 |
|  288  |15.8678 |  1   |   0.000232   |  48.548550   |   0.000000   | 0.000000e+00 | 0.000000e+00 |
|  289  |15.9229 |  1   |   0.000224   |  51.276163   |   0.000000   | 0.000000e+00 | 0.000000e+00 |
|  290  |15.9780 |  1   |   0.000233   |  48.570122   |   0.000000   | 0.000000e+00 | 0.000000e+00 |
|  291  |16.0331 |  1   |   0.000192   |  50.266936   |   0.000000   | 0.000000e+00 | 0.000000e+00 |
|  292  |16.0882 |  1   |   0.000295   |  48.692444   |   0.000000   | 0.000000e+00 | 0.000000e+00 |
|  293  |16.1433 |  1   |   0.000202   |  48.893389   |   0.000000   | 0.000000e+00 | 0.000000e+00 |
|  294  |16.1983 |  1   |   0.000443   |  50.813372   |   0.000000   | 0.000000e+00 | 0.000000e+00 |
|  295  |16.2534 |  1   |   0.000529   |  60.601432   |   0.000000   | 0.000000e+00 | 0.000000e+00 |
|  296  |16.3085 |  1   |   0.000571   |  50.993108   |   0.000000   | 0.000000e+00 | 0.000000e+00 |
|  297  |16.3636 |  1   |   0.000228   |  49.065097   |   0.000000   | 0.000000e+00 | 0.000000e+00 |
|  298  |16.4187 |  1   |   0.000234   |  48.226984   |   0.000000   | 0.000000e+00 | 0.000000e+00 |
|  299  |16.4738 |  1   |   0.000233   |  48.534926   |   0.000000   | 0.000000e+00 | 0.000000e+00 |
|  300  |16.5289 |  1   |   0.000627   |  48.788802   |   0.000000   | 0.000000e+00 | 0.000000e+00 |
|  301  |16.5840 |  1   |   0.000555   |  60.530895   |   0.000000   | 0.000000e+00 | 0.000000e+00 |
|  302  |16.6391 |  1   |   0.000628   |  48.944436   |   0.000000   | 0.000000e+00 | 0.000000e+00 |
|  303  |16.6942 |  1   |   0.000427   |  49.049230   |   0.000000   | 0.000000e+00 | 0.000000e+00 |
|  304  |16.7493 |  1   |   0.000236   |  47.892993   |   0.000000   | 0.000000e+00 | 0.000000e+00 |
|  305  |16.8044 |  1   |   0.000325   |  48.441506   |   0.000000   | 0.000000e+00 | 0.000000e+00 |
|  306  |16.8595 |  1   |   0.000236   |  48.047404   |   0.000000   | 0.000000e+00 | 0.000000e+00 |
|  307  |16.9146 |  1   |   0.000232   |  48.215284   |   0.000000   | 0.000000e+00 | 0.000000e+00 |
|  308  |16.9697 |  1   |   0.000226   |  50.124858   |   0.000000   | 0.000000e+00 | 0.000000e+00 |
|  309  |17.0248 |  1   |   0.000213   |  52.395805   |   0.000000   | 0.000000e+00 | 0.000000e+00 |
|  310  |17.0799 |  1   |   0.000222   |  51.302776   |   0.000000   | 0.000000e+00 | 0.000000e+00 |
|  311  |17.1350 |  1   |   0.000439   |  51.171773   |   0.000000   | 0.000000e+00 | 0.000000e+00 |
|  312  |17.1901 |  1   |   0.000232   |  49.032143   |   0.000000   | 0.000000e+00 | 0.000000e+00 |
|  313  |17.2452 |  1   |   0.000233   |  48.628400   |   0.000000   | 0.000000e+00 | 0.000000e+00 |
|  314  |17.3003 |  1   |   0.000237   |  47.940311   |   0.000000   | 0.000000e+00 | 0.000000e+00 |
|  315  |17.3554 |  1   |   0.000216   |  51.865635   |   0.000000   | 0.000000e+00 | 0.000000e+00 |
|  316  |17.4105 |  1   |   0.000373   |  54.905208   |   0.000000   | 0.000000e+00 | 0.000000e+00 |
|  317  |17.4656 |  1   |   0.000310   |  48.191338   |   0.000000   | 0.000000e+00 | 0.000000e+00 |
|  318  |17.5207 |  1   |   0.000409   |  48.607705   |   0.000000   | 0.000000e+00 | 0.000000e+00 |
|  319  |17.5758 |  1   |   0.000350   |  53.126807   |   0.000000   | 0.000000e+00 | 0.000000e+00 |
|  320  |17.6309 |  1   |   0.000236   |  47.827493   |   0.000000   | 0.000000e+00 | 0.000000e+00 |
|  321  |17.6860 |  1   |   0.000233   |  48.142512   |   0.000000   | 0.000000e+00 | 0.000000e+00 |
|  322  |17.7410 |  1   |   0.000233   |  48.315800   |   0.000000   | 0.000000e+00 | 0.000000e+00 |
|  323  |17.7961 |  1   |   0.000203   |  55.821999   |   0.000000   | 0.000000e+00 | 0.000000e+00 |
|  324  |17.8512 |  1   |   0.000227   |  49.786573   |   0.000000   | 0.000000e+00 | 0.000000e+00 |
|  325  |17.9063 |  1   |   0.000424   |  53.387859   |   0.000000   | 0.000000e+00 | 0.000000e+00 |
|  326  |17.9614 |  1   |   0.000447   |  50.685506   |   0.000000   | 0.000000e+00 | 0.000000e+00 |
|  327  |18.0165 |  1   |   0.000235   |  48.078798   |   0.000000   | 0.000000e+00 | 0.000000e+00 |
|  328  |18.0716 |  1   |   0.000232   |  48.322922   |   0.000000   | 0.000000e+00 | 0.000000e+00 |
|  329  |18.1267 |  1   |   0.000223   |  48.098878   |   0.000000   | 0.000000e+00 | 0.000000e+00 |
|  330  |18.1818 |  1   |   0.000201   |  48.656743   |   0.000000   | 0.000000e+00 | 0.000000e+00 |
|  331  |18.2369 |  1   |   0.000273   |  48.314217   |   0.000000   | 0.000000e+00 | 0.000000e+00 |
|  332  |18.2920 |  1   |   0.000224   |  48.475068   |   0.000000   | 0.000000e+00 | 0.000000e+00 |
|  333  |18.3471 |  1   |   0.000230   |  48.642246   |   0.000000   | 0.000000e+00 | 0.000000e+00 |
|  334  |18.4022 |  1   |   0.000194   |  57.654265   |   0.000000   | 0.000000e+00 | 0.000000e+00 |
|  335  |18.4573 |  1   |   0.000225   |  50.092508   |   0.000000   | 0.000000e+00 | 0.000000e+00 |
|  336  |18.5124 |  1   |   0.000291   |  49.505815   |   0.000000   | 0.000000e+00 | 0.000000e+00 |
|  337  |18.5675 |  1   |   0.000228   |  49.799143   |   0.000000   | 0.000000e+00 | 0.000000e+00 |
|  338  |18.6226 |  1   |   0.000272   |  48.344175   |   0.000000   | 0.000000e+00 | 0.000000e+00 |
|  339  |18.6777 |  1   |   0.000219   |  48.669889   |   0.000000   | 0.000000e+00 | 0.000000e+00 |
|  340  |18.7328 |  1   |   0.000221   |  48.424671   |   0.000000   | 0.000000e+00 | 0.000000e+00 |
|  341  |18.7879 |  1   |   0.000204   |  55.293948   |   0.000000   | 0.000000e+00 | 0.000000e+00 |
|  342  |18.8430 |  1   |   0.000361   |  48.517776   |   0.000000   | 0.000000e+00 | 0.000000e+00 |
|  343  |18.8981 |  1   |   0.000218   |  52.256841   |   0.000000   | 0.000000e+00 | 0.000000e+00 |
|  344  |18.9532 |  1   |   0.000232   |  48.441100   |   0.000000   | 0.000000e+00 | 0.000000e+00 |
|  345  |19.0083 |  1   |   0.000234   |  48.389977   |   0.000000   | 0.000000e+00 | 0.000000e+00 |
|  346  |19.0634 |  1   |   0.000230   |  48.982534   |   0.000000   | 0.000000e+00 | 0.000000e+00 |
|  347  |19.1185 |  1   |   0.000231   |  48.746604   |   0.000000   | 0.000000e+00 | 0.000000e+00 |
|  348  |19.1736 |  1   |   0.000229   |  56.961390   |   0.000000   | 0.000000e+00 | 0.000000e+00 |
|  349  |19.2287 |  1   |   0.000188   |  59.830517   |   0.000000   | 0.000000e+00 | 0.000000e+00 |
|  350  |19.2837 |  1   |   0.000296   |  51.906979   |   0.000000   | 0.000000e+00 | 0.000000e+00 |
|  351  |19.3388 |  1   |   0.000217   |  51.754677   |   0.000000   | 0.000000e+00 | 0.000000e+00 |
|  352  |19.3939 |  1   |   0.000244   |  48.090983   |   0.000000   | 0.000000e+00 | 0.000000e+00 |
|  353  |19.4490 |  1   |   0.000234   |  48.219884   |   0.000000   | 0.000000e+00 | 0.000000e+00 |
|  354  |19.5041 |  1   |   0.000211   |  53.274204   |   0.000000   | 0.000000e+00 | 0.000000e+00 |
|  355  |19.5592 |  1   |   0.000236   |  47.988238   |   0.000000   | 0.000000e+00 | 0.000000e+00 |
|  356  |19.6143 |  1   |   0.000236   |  48.149312   |   0.000000   | 0.000000e+00 | 0.000000e+00 |
|  357  |19.6694 |  1   |   0.000231   |  48.684465   |   0.000000   | 0.000000e+00 | 0.000000e+00 |
|  358  |19.7245 |  1   |   0.000245   |  48.573752   |   0.000000   | 0.000000e+00 | 0.000000e+00 |
|  359  |19.7796 |  1   |   0.000255   |  48.387298   |   0.000000   | 0.000000e+00 | 0.000000e+00 |
|  360  |19.8347 |  1   |   0.000233   |  48.496933   |   0.000000   | 0.000000e+00 | 0.000000e+00 |
|  361  |19.8898 |  1   |   0.000298   |  48.227055   |   0.000000   | 0.000000e+00 | 0.000000e+00 |
|  362  |19.9449 |  1   |   0.000277   |  50.033512   |   0.000000   | 0.000000e+00 | 0.000000e+00 |
|  363  |20.0000 |  1   |   0.000230   |  48.865905   |   0.000000   | 0.000000e+00 | 0.000000e+00 |
|  364  |20.0551 |  1   |   0.000233   |  48.282476   |   0.000000   | 0.000000e+00 | 0.000000e+00 |
|  365  |20.1102 |  1   |   0.000231   |  48.869913   |   0.000000   | 0.000000e+00 | 0.000000e+00 |
|  366  |20.1653 |  1   |   0.000240   |  48.197653   |   0.000000   | 0.000000e+00 | 0.000000e+00 |
|  367  |20.2204 |  1   |   0.000236   |  47.995288   |   0.000000   | 0.000000e+00 | 0.000000e+00 |
|  368  |20.2755 |  1   |   0.000218   |  48.416211   |   0.000000   | 0.000000e+00 | 0.000000e+00 |
|  369  |20.3306 |  1   |   0.000229   |  49.650859   |   0.000000   | 0.000000e+00 | 0.000000e+00 |
|  370  |20.3857 |  1   |   0.000281   |  50.925766   |   0.000000   | 0.000000e+00 | 0.000000e+00 |
|  371  |20.4408 |  1   |   0.000293   |  49.072721   |   0.000000   | 0.000000e+00 | 0.000000e+00 |
|  372  |20.4959 |  1   |   0.000237   |  47.848434   |   0.000000   | 0.000000e+00 | 0.000000e+00 |
|  373  |20.5510 |  1   |   0.000461   |  48.577324   |   0.000000   | 0.000000e+00 | 0.000000e+00 |
|  374  |20.6061 |  1   |   0.000288   |  48.149040   |   0.000000   | 0.000000e+00 | 0.000000e+00 |
|  375  |20.6612 |  1   |   0.000232   |  48.140545   |   0.000000   | 0.000000e+00 | 0.000000e+00 |
|  376  |20.7163 |  1   |   0.000627   |  48.528365   |   0.000000   | 0.000000e+00 | 0.000000e+00 |
|  377  |20.7713 |  1   |   0.000649   |  48.815492   |   0.000000   | 0.000000e+00 | 0.000000e+00 |
|  378  |20.8264 |  1   |   0.000220   |  48.691541   |   0.000000   | 0.000000e+00 | 0.000000e+00 |
|  379  |20.8815 |  1   |   0.000234   |  48.163874   |   0.000000   | 0.000000e+00 | 0.000000e+00 |
|  380  |20.9366 |  1   |   0.000397   |  47.970319   |   0.000000   | 0.000000e+00 | 0.000000e+00 |
|  381  |20.9917 |  1   |   0.000266   |  49.183593   |   0.000000   | 0.000000e+00 | 0.000000e+00 |
|  382  |21.0468 |  1   |   0.000228   |  49.448159   |   0.000000   | 0.000000e+00 | 0.000000e+00 |
|  383  |21.1019 |  1   |   0.000230   |  48.558385   |   0.000000   | 0.000000e+00 | 0.000000e+00 |
|  384  |21.1570 |  1   |   0.000279   |  48.382636   |   0.000000   | 0.000000e+00 | 0.000000e+00 |
|  385  |21.2121 |  1   |   0.000237   |  47.482272   |   0.000000   | 0.000000e+00 | 0.000000e+00 |
|  386  |21.2672 |  1   |   0.000233   |  48.183362   |   0.000000   | 0.000000e+00 | 0.000000e+00 |
|  387  |21.3223 |  1   |   0.000232   |  48.062299   |   0.000000   | 0.000000e+00 | 0.000000e+00 |
|  388  |21.3774 |  1   |   0.000235   |  48.362948   |   0.000000   | 0.000000e+00 | 0.000000e+00 |
|  389  |21.4325 |  1   |   0.000233   |  48.422455   |   0.000000   | 0.000000e+00 | 0.000000e+00 |
|  390  |21.4876 |  1   |   0.000236   |  48.077169   |   0.000000   | 0.000000e+00 | 0.000000e+00 |
|  391  |21.5427 |  1   |   0.000218   |  48.317787   |   0.000000   | 0.000000e+00 | 0.000000e+00 |
|  392  |21.5978 |  1   |   0.000231   |  48.382455   |   0.000000   | 0.000000e+00 | 0.000000e+00 |
|  393  |21.6529 |  1   |   0.000233   |  48.310206   |   0.000000   | 0.000000e+00 | 0.000000e+00 |
|  394  |21.7080 |  1   |   0.000219   |  49.225511   |   0.000000   | 0.000000e+00 | 0.000000e+00 |
|  395  |21.7631 |  1   |   0.000238   |  47.725182   |   0.000000   | 0.000000e+00 | 0.000000e+00 |
|  396  |21.8182 |  1   |   0.000403   |  48.100903   |   0.000000   | 0.000000e+00 | 0.000000e+00 |
|  397  |21.8733 |  1   |   0.000227   |  49.886770   |   0.000000   | 0.000000e+00 | 0.000000e+00 |
|  398  |21.9284 |  1   |   0.000226   |  49.658201   |   0.000000   | 0.000000e+00 | 0.000000e+00 |
|  399  |21.9835 |  1   |   0.000236   |  48.305113   |   0.000000   | 0.000000e+00 | 0.000000e+00 |
|  400  |22.0386 |  1   |   0.000213   |  53.014337   |   0.000000   | 0.000000e+00 | 0.000000e+00 |
|  401  |22.0937 |  1   |   0.000323   |  48.752367   |   0.000000   | 0.000000e+00 | 0.000000e+00 |
|  402  |22.1488 |  1   |   0.000231   |  48.655110   |   0.000000   | 0.000000e+00 | 0.000000e+00 |
|  403  |22.2039 |  1   |   0.000302   |  49.894845   |   0.000000   | 0.000000e+00 | 0.000000e+00 |
|  404  |22.2590 |  1   |   0.000192   |  54.227928   |   0.000000   | 0.000000e+00 | 0.000000e+00 |
|  405  |22.3140 |  1   |   0.000213   |  53.815790   |   0.000000   | 0.000000e+00 | 0.000000e+00 |
|  406  |22.3691 |  1   |   0.000231   |  49.235416   |   0.000000   | 0.000000e+00 | 0.000000e+00 |
|  407  |22.4242 |  1   |   0.000232   |  48.387503   |   0.000000   | 0.000000e+00 | 0.000000e+00 |
|  408  |22.4793 |  1   |   0.000220   |  52.815450   |   0.000000   | 0.000000e+00 | 0.000000e+00 |
|  409  |22.5344 |  1   |   0.000232   |  48.253041   |   0.000000   | 0.000000e+00 | 0.000000e+00 |
|  410  |22.5895 |  1   |   0.000218   |  52.584628   |   0.000000   | 0.000000e+00 | 0.000000e+00 |
|  411  |22.6446 |  1   |   0.000222   |  50.874607   |   0.000000   | 0.000000e+00 | 0.000000e+00 |
|  412  |22.6997 |  1   |   0.000234   |  48.125687   |   0.000000   | 0.000000e+00 | 0.000000e+00 |
|  413  |22.7548 |  1   |   0.000314   |  65.912785   |   0.000000   | 0.000000e+00 | 0.000000e+00 |
|  414  |22.8099 |  1   |   0.000294   |  48.712801   |   0.000000   | 0.000000e+00 | 0.000000e+00 |
|  415  |22.8650 |  1   |   0.000234   |  48.264049   |   0.000000   | 0.000000e+00 | 0.000000e+00 |
|  416  |22.9201 |  1   |   0.000219   |  52.101943   |   0.000000   | 0.000000e+00 | 0.000000e+00 |
|  417  |22.9752 |  1   |   0.000233   |  48.181907   |   0.000000   | 0.000000e+00 | 0.000000e+00 |
|  418  |23.0303 |  1   |   0.000239   |  47.737343   |   0.000000   | 0.000000e+00 | 0.000000e+00 |
|  419  |23.0854 |  1   |   0.000189   |  55.077763   |   0.000000   | 0.000000e+00 | 0.000000e+00 |
|  420  |23.1405 |  1   |   0.000252   |  48.412007   |   0.000000   | 0.000000e+00 | 0.000000e+00 |
|  421  |23.1956 |  1   |   0.000207   |  52.025445   |   0.000000   | 0.000000e+00 | 0.000000e+00 |
|  422  |23.2507 |  1   |   0.000202   |  55.306192   |   0.000000   | 0.000000e+00 | 0.000000e+00 |
|  423  |23.3058 |  1   |   0.000235   |  48.069254   |   0.000000   | 0.000000e+00 | 0.000000e+00 |
|  424  |23.3609 |  1   |   0.000268   |  48.786326   |   0.000000   | 0.000000e+00 | 0.000000e+00 |
|  425  |23.4160 |  1   |   0.000234   |  48.662434   |   0.000000   | 0.000000e+00 | 0.000000e+00 |
|  426  |23.4711 |  1   |   0.000430   |  48.229208   |   0.000000   | 0.000000e+00 | 0.000000e+00 |
|  427  |23.5262 |  1   |   0.000660   |  48.355178   |   0.000000   | 0.000000e+00 | 0.000000e+00 |
|  428  |23.5813 |  1   |   0.000475   |  47.940589   |   0.000000   | 0.000000e+00 | 0.000000e+00 |
|  429  |23.6364 |  1   |   0.000464   |  48.627356   |   0.000000   | 0.000000e+00 | 0.000000e+00 |
|  430  |23.6915 |  1   |   0.000231   |  48.519762   |   0.000000   | 0.000000e+00 | 0.000000e+00 |
|  431  |23.7466 |  1   |   0.000233   |  48.359547   |   0.000000   | 0.000000e+00 | 0.000000e+00 |
|  432  |23.8017 |  1   |   0.000273   |  48.313746   |   0.000000   | 0.000000e+00 | 0.000000e+00 |
|  433  |23.8567 |  1   |   0.000237   |  47.977213   |   0.000000   | 0.000000e+00 | 0.000000e+00 |
|  434  |23.9118 |  1   |   0.000226   |  49.786822   |   0.000000   | 0.000000e+00 | 0.000000e+00 |
|  435  |23.9669 |  1   |   0.000231   |  48.641675   |   0.000000   | 0.000000e+00 | 0.000000e+00 |
|  436  |24.0220 |  1   |   0.000263   |  50.056180   |   0.000000   | 0.000000e+00 | 0.000000e+00 |
|  437  |24.0771 |  1   |   0.000239   |  48.169318   |   0.000000   | 0.000000e+00 | 0.000000e+00 |
|  438  |24.1322 |  1   |   0.000237   |  47.903935   |   0.000000   | 0.000000e+00 | 0.000000e+00 |
|  439  |24.1873 |  1   |   0.000235   |  48.071315   |   0.000000   | 0.000000e+00 | 0.000000e+00 |
|  440  |24.2424 |  1   |   0.000298   |  47.997039   |   0.000000   | 0.000000e+00 | 0.000000e+00 |
|  441  |24.2975 |  1   |   0.000204   |  51.404477   |   0.000000   | 0.000000e+00 | 0.000000e+00 |
|  442  |24.3526 |  1   |   0.000234   |  48.367934   |   0.000000   | 0.000000e+00 | 0.000000e+00 |
|  443  |24.4077 |  1   |   0.000297   |  48.626223   |   0.000000   | 0.000000e+00 | 0.000000e+00 |
|  444  |24.4628 |  1   |   0.000254   |  48.214660   |   0.000000   | 0.000000e+00 | 0.000000e+00 |
|  445  |24.5179 |  1   |   0.000234   |  47.731147   |   0.000000   | 0.000000e+00 | 0.000000e+00 |
|  446  |24.5730 |  1   |   0.000268   |  48.701918   |   0.000000   | 0.000000e+00 | 0.000000e+00 |
|  447  |24.6281 |  1   |   0.000233   |  48.425494   |   0.000000   | 0.000000e+00 | 0.000000e+00 |
|  448  |24.6832 |  1   |   0.000233   |  48.324203   |   0.000000   | 0.000000e+00 | 0.000000e+00 |
|  449  |24.7383 |  1   |   0.000228   |  49.378323   |   0.000000   | 0.000000e+00 | 0.000000e+00 |
|  450  |24.7934 |  1   |   0.000229   |  48.777402   |   0.000000   | 0.000000e+00 | 0.000000e+00 |
...Finished
FINISHED - Elapsed time = 23245.3692466 seconds
FINISHED - CPU process time = 92349.9106726 seconds

Postprocessing¶

Read the structural and aerodynamic information of the last time step

[11]:

tstep = sharpy_output.structure.timestep_info[-1]
astep = sharpy_output.aero.timestep_info[-1]

Separate the structure into blades

[12]:

# Define beams
ielem = 0
nblades = np.max(sharpy_output.structure.beam_number) + 1
nodes_blade = []
first_node = 0
for iblade in range(nblades):
    nodes_blade.append(np.zeros((sharpy_output.structure.num_node,), dtype=bool))
    while sharpy_output.structure.beam_number[ielem] <= iblade:
        ielem += 1
        if ielem == sharpy_output.structure.num_elem:
            break
    nodes_blade[iblade][first_node:sharpy_output.structure.connectivities[ielem-1,1]+1] = True
    first_node = sharpy_output.structure.connectivities[ielem-1,1]+1

Compute the radial position of the nodes and initialise the rest of the variables

[13]:

r = []
c = []
dr = []
forces = []
CN_drR = []
CTan_drR = []
CP_drR = []
nodes_num = []
for iblade in range(nblades):
    forces.append(tstep.steady_applied_forces[nodes_blade[iblade]].copy())

    nodes_num.append(np.arange(0, sharpy_output.structure.num_node, 1)[nodes_blade[iblade]])

    r.append(np.linalg.norm(tstep.pos[nodes_blade[iblade], :], axis=1))
    dr.append(np.zeros(np.sum(nodes_blade[iblade])))
    dr[iblade][0] = 0.5*(r[iblade][1]-r[iblade][0])
    dr[iblade][-1] = 0.5 * (r[iblade][-1] - r[iblade][-2])
    for inode in range(1,len(r[iblade]) - 1):
        dr[iblade][inode] = 0.5*(r[iblade][inode+1] - r[iblade][inode-1])

    CN_drR.append(np.zeros(len(r[iblade])))
    c.append(np.zeros(len(r[iblade])))
    CTan_drR.append(np.zeros(len(r[iblade])))
    CP_drR.append(np.zeros(len(r[iblade])))

Transform the loads computed by SHARPy into out-of-plane and in-plane components

[14]:

rho = sharpy_output.settings['StaticCoupledRBM']['aero_solver_settings']['rho']
uinf = sharpy_output.settings['StaticCoupledRBM']['aero_solver_settings']['velocity_field_input']['u_inf']
R = np.max(r[0])
Cp = 0
Ct = 0

global_force_factor = 0.5 * rho * uinf** 2 * np.pi * R**2
global_power_factor = global_force_factor*uinf
for iblade in range(nblades):
    for inode in range(len(r[iblade])):
        forces[iblade][inode, 0] *= 0. # Discard the spanwise component

        node_global_index = nodes_num[iblade][inode]
        ielem = sharpy_output.structure.node_master_elem[node_global_index, 0]
        inode_in_elem = sharpy_output.structure.node_master_elem[node_global_index, 1]
        CAB = algebra.crv2rotation(tstep.psi[ielem, inode_in_elem, :])

        c[iblade][inode] = sharpy_output.aero.aero_dict['chord'][ielem,inode_in_elem]

        forces_AFoR = np.dot(CAB, forces[iblade][inode, 0:3])

        CN_drR[iblade][inode] = forces_AFoR[2]/dr[iblade][inode]*R / global_force_factor
        CTan_drR[iblade][inode] = np.linalg.norm(forces_AFoR[0:2])/dr[iblade][inode]*R  / global_force_factor
        CP_drR[iblade][inode] = np.linalg.norm(forces_AFoR[0:2])/dr[iblade][inode]*R  * r[iblade][inode]*rotation_velocity / global_power_factor

    Cp += np.sum(CP_drR[iblade]*dr[iblade]/R)
    Ct += np.sum(CN_drR[iblade]*dr[iblade]/R)

Results¶

Plot of the loads along the blade

[15]:

fig, list_plots = plt.subplots(1, 2, figsize=(12, 3))

list_plots[0].grid()
list_plots[0].set_xlabel("r/R [-]")
list_plots[0].set_ylabel("CN/d(r/R) [-]")
list_plots[0].plot(r[0]/R, CN_drR[0], '-', label='SHARPy')
list_plots[0].plot(of_rR, of_cNdrR, '-', label='OpenFAST')
list_plots[0].legend()

list_plots[1].grid()
list_plots[1].set_xlabel("r/R [-]")
list_plots[1].set_ylabel("CT/d(r/R) [-]")
list_plots[1].plot(r[0]/R, CTan_drR[0], '-', label='SHARPy')
list_plots[1].plot(of_rR, of_cTdrR, '-', label='OpenFAST')
list_plots[1].legend()

plt.show()

_images/content_example_notebooks_wind_turbine_34_0.svg

Print the rotor thrust and power coefficients

[16]:

print("      OpenFAST SHARPy")
print("Cp[-] %.2f       %.2f" % (of_cp, Cp))
print("Ct[-] %.2f       %.2f" % (of_ct, Ct))

      OpenFAST SHARPy
Cp[-] 0.49       0.52
Ct[-] 0.70       0.70

A prismatic cantilever beam under a tip load¶

This is an example of the geometrically-exact beam structural solver in SHARPy.

Reference: Simpson R.J.S., Palacios R., “Numerical Aspects of Nonlinear Flexible Aircraft Flight Dynamics Modeling.” 54th AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics and Materials Conference, 8-11 April 2013, Boston, Massachusetts, USA [http://hdl.handle.net/10044/1/11077, http://dx.doi.org/10.2514/6.2013-1634]

Required Packages¶

[1]:

import numpy as np
import os
import matplotlib.pyplot as plt
import sharpy.sharpy_main                # used to run SHARPy from Jupyter
import model_static_cantilever as model  # model definition
from IPython.display import Image

plt.rcParams.update({'font.size': 20})   # Large fonts in all plots

Problem 1: Tip vertical dead load¶

Consider first a massless beam with a heavy tip mass, such that the deformations are due to the resulting dead load P. The static equilibrium will be obtained for multiple values of P=0, 100, …, 1000 kN

[2]:

Image('images/cantilever.png', width=500)

[2]:

_images/content_example_notebooks_cantilever_static_cantilever_4_0.png

[3]:

# Define temporary files to generate sharpy models.
case_name= 'temp'
route = './'

Nforces=10         # Number of force steps
DeltaForce=100e3 # Increment of forces
Nelem=20          # Number of beam elements

N=2*Nelem+1
x1=np.zeros((Nforces,N))
z1=np.zeros((Nforces,N))

#Loop through all external forces
for jForce in range(Nforces):
    model.clean_test_files(route, case_name)
    model.generate_fem_file(route, case_name, Nelem, float(jForce+1)*DeltaForce)
    model.generate_solver_file(route,case_name)

    case_data=sharpy.sharpy_main.main(['', route + case_name + '.sharpy'])

    x1[jForce,0:N]=case_data.structure.timestep_info[0].pos[:, 0]
    z1[jForce,0:N]=case_data.structure.timestep_info[0].pos[:, 2]

#Store initial geometry
x0=case_data.structure.ini_info.pos[:, 0]
z0=case_data.structure.ini_info.pos[:, 2]

[4]:

# Plot the deformed beam shapes
fig= plt.subplots(1, 1, figsize=(15, 6))
plt.scatter(x0,z0,c='black')
plt.plot(x0,z0,c='black')

for jForce in range(Nforces):
    plt.scatter(x1[jForce,0:N],z1[jForce,0:N],c='black')
    plt.plot(x1[jForce,0:N],z1[jForce,0:N],c='black')

plt.axis('equal')
plt.grid()
plt.xlabel('x (m)')
plt.ylabel('z (m)')
plt.savefig("images/ncb1-dead-displ.eps",  format='eps', dpi=1000, bbox_inches='tight')

_images/content_example_notebooks_cantilever_static_cantilever_6_0.png

[5]:

print('{:>8s}{:>12s}'.format('Force','Tip z'))
dash=20*'-'; print(dash)
for jForce in range(Nforces):
    print('{:>8.0f}{:>12.4f}'.format((jForce+1)*DeltaForce,z1[jForce,N-1]))

   Force       Tip z
--------------------
  100000     -0.4438
  200000     -0.8672
  300000     -1.2551
  400000     -1.5999
  500000     -1.9005
  600000     -2.1597
  700000     -2.3824
  800000     -2.5737
  900000     -2.7386
 1000000     -2.8813

[6]:

model.clean_test_files(route, case_name)

Problem 2: Comparing follower and dead forces¶

[7]:

#Loop through all external follower forces, again applied at the tip.
x2=np.zeros((Nforces,N))
z2=np.zeros((Nforces,N))
for jForce in range(Nforces):
    model.clean_test_files(route, case_name)
    model.generate_fem_file(route, case_name, Nelem, 0, -float(jForce+1)*DeltaForce)
    model.generate_solver_file(route,case_name)

    case_foll=sharpy.sharpy_main.main(['', route + case_name + '.sharpy'])

    x2[jForce,0:N]=case_foll.structure.timestep_info[0].pos[:, 0]
    z2[jForce,0:N]=case_foll.structure.timestep_info[0].pos[:, 2]

[8]:

#Plot results.
fig, axs = plt.subplots(nrows=1, ncols=2, figsize=(15, 6))

ax0 = axs[0]
ax0.plot([0,Nforces],[0, 0],linestyle=':', c='black')
ax0.plot(range(Nforces+1), np.concatenate((5, x1[0:Nforces,-1]), axis=None)-5*np.ones(Nforces+1),linestyle='-', c='black')
ax0.plot(range(Nforces+1), np.concatenate((5, x2[0:Nforces,-1]), axis=None)-5*np.ones(Nforces+1),linestyle='--', c='black')
#ax0.axis('equal')
ax0.grid()
ax0.set_xlabel('Force (N) x 10^5')
ax0.set_ylabel('Tip horizontal displacement (m)')
ax0.set(xlim=(0, Nforces), ylim=(-6, 1))

ax1 = axs[1]
ax1.plot([0,Nforces],np.concatenate((0, z1[0,-1]*Nforces), axis=None),linestyle=':', c='black')
ax1.plot(range(Nforces+1), np.concatenate((0, z1[0:Nforces,-1]), axis=None),linestyle='-', c='black')
ax1.plot(range(Nforces+1), np.concatenate((0, z2[0:Nforces,-1]), axis=None),linestyle='--', c='black')


#ax1.axis('equal')
ax1.grid()
ax1.set_xlabel('Force (N) x 10^5')
ax1.set_ylabel('Tip vertical displacement (m)')
ax1.set(xlim=(0, Nforces), ylim=(-5, 1))

ax1.legend(['Linear','Dead','Follower'])

fig.savefig("images/ncb1-foll-displ.eps", format='eps', dpi=1000, bbox_inches="tight")

The PostScript backend does not support transparency; partially transparent artists will be rendered opaque.
The PostScript backend does not support transparency; partially transparent artists will be rendered opaque.
The PostScript backend does not support transparency; partially transparent artists will be rendered opaque.
The PostScript backend does not support transparency; partially transparent artists will be rendered opaque.

_images/content_example_notebooks_cantilever_static_cantilever_11_1.png

[9]:

model.clean_test_files(route, case_name)

A first test case with SHARPy¶

This notebook explains how to create a prismatic cantilever and flexible wing in SHARPy. It aims to provide a big picture about the simulations available in SHARPy describing the very fundamental concepts.

This notebook requires around two hours to be completed.

Generation of the aeroelastic model¶

This section explains how to generate the input files for SHARPy including the structure, the aerodynamics and the simulation details.

[3]:

# Loading of the used packages
import numpy as np              # basic mathematical and array functions
import os                       # Functions related to the operating system
import matplotlib.pyplot as plt # Plotting library

import sharpy.sharpy_main                  # Run SHARPy inside jupyter notebooks
import sharpy.utils.plotutils as pu        # Plotting utilities
from sharpy.utils.constants import deg2rad # Constant to conver degrees to radians

import sharpy.utils.generate_cases as gc

The generate cases module of SHARPy includes a number of templates for the creation of simple cases using common parameters as inputs. It also removes clutter from this notebook.

The following cell configures the plotting library to show the next graphics, it is not part of SHARPy.

[4]:

%%capture
! pip install plotly

Define the problem parameters¶

In this section, we define the basic parameters to generate a model for the wing shown in the image below. We use SI units for all variables.

[5]:

%config InlineBackend.figure_format = 'svg'
from IPython.display import Image
url = ('https://raw.githubusercontent.com/ImperialCollegeLondon/sharpy/dev_example/docs/' +
       'source/content/example_notebooks/images/simple_wing_scheme.png')
Image(url=url, width=800)

[5]:

[6]:

# Geometry
chord = 1.         # Chord of the wing
aspect_ratio = 16. # Ratio between lenght and chord: aspect_ratio = length/chord
wake_length = 50   # Length of the wake in chord lengths

# Discretization
num_node = 21           # Number of nodes in the structural discretisation
                        # The number of nodes will also define the aerodynamic panels in the
                        # spanwise direction
num_chord_panels = 4    # Number of aerodynamic panels in the chordwise direction
num_points_camber = 200 # The camber line of the wing will be defined by a series of (x,y)
                        # coordintes. Here, we define the size of the (x,y) vectors

# Structural properties of the beam cross section
mass_per_unit_length = 0.75 # Mass per unit length
mass_iner_x = 0.1           # Mass inertia around the local x axis
mass_iner_y = 0.05          # Mass inertia around the local y axis
mass_iner_z = 0.05          # Mass inertia around the local z axis
pos_cg_B = np.zeros((3))    # position of the centre of mass with respect to the elastic axis
EA = 1e7                    # Axial stiffness
GAy = 1e6                   # Shear stiffness in the local y axis
GAz = 1e6                   # Shear stiffness in the local z axis
GJ = 1e4                    # Torsional stiffness
EIy = 2e4                   # Bending stiffness around the flapwise direction
EIz = 5e6                   # Bending stiffness around the edgewise direction

# Operation
WSP = 2.                # Wind speed
air_density = 0.1       # Air density

# Time discretization
end_time = 5.0                  # End time of the simulation
dt = chord/num_chord_panels/WSP # Always keep one timestep per panel

First, we are going to compute the static equilibrium of the wing at an angle of attack (aoa_ini_deg). Next, we will change the angle of attack to aoa_end_deg and we will compute the dynamic response of the wing.

[7]:

aoa_ini_deg = 2.        # Angle of attack at the beginning of the simulation
aoa_end_deg = 1.        # Angle of attack at the end of the simulation

For now, we keep the size of the wake panels equal to the distance covered by the flow in one time step.

Structural model¶

This section creates a class called AeroelasticInformation. This class is not directly used by SHARPy, it has been thought as an intermediate step between common engineering inputs and the input information that SHARPy needs.

It has many functionalities (rotate objects, assembly simple strucures together …) which can be looked up in the documentation.

Let’s initialise an Aeroelastic system that will include a StructuralInformation class and an AerodynamicInformation class:

[8]:

wing = gc.AeroelasticInformation()

The attibutes that have to be defined (not all of them are compulsory) can be displayed with the code below. They constitute the input parameters to SHARPy and their names are intuitive, however, a complete description of the input variables to SHARPy can be found in the documentation: structural inputs and aerodynamic inputs

For example, the structural properties are:

[9]:

wing.StructuralInformation.__dict__.keys()

[9]:

dict_keys(['num_node_elem', 'num_node', 'num_elem', 'coordinates', 'connectivities', 'elem_stiffness', 'stiffness_db', 'elem_mass', 'mass_db', 'frame_of_reference_delta', 'structural_twist', 'boundary_conditions', 'beam_number', 'body_number', 'app_forces', 'lumped_mass_nodes', 'lumped_mass', 'lumped_mass_inertia', 'lumped_mass_position', 'lumped_mass_mat', 'lumped_mass_mat_nodes'])

For example, the connectivities between the nodes required by the finite element solver are empty so far:

[10]:

print(wing.StructuralInformation.connectivities)

None

For a list of methods that allow us to modify the structure, Check the documentation.

First, we need to define the basic characteristics of the equations as the number of nodes, the number of nodes per element, the number of elements and the location of the nodes in the space:

[11]:

# Define the number of nodes and the number of nodes per element
wing.StructuralInformation.num_node = num_node
wing.StructuralInformation.num_node_elem = 3
# Compute the number of elements assuming basic connections
wing.StructuralInformation.compute_basic_num_elem()

[12]:

# Generate an array with the location of the nodes
node_r = np.zeros((num_node, 3))
node_r[:,1] = np.linspace(0, chord*aspect_ratio, num_node)
print(node_r)

[[ 0.   0.   0. ]
 [ 0.   0.8  0. ]
 [ 0.   1.6  0. ]
 [ 0.   2.4  0. ]
 [ 0.   3.2  0. ]
 [ 0.   4.   0. ]
 [ 0.   4.8  0. ]
 [ 0.   5.6  0. ]
 [ 0.   6.4  0. ]
 [ 0.   7.2  0. ]
 [ 0.   8.   0. ]
 [ 0.   8.8  0. ]
 [ 0.   9.6  0. ]
 [ 0.  10.4  0. ]
 [ 0.  11.2  0. ]
 [ 0.  12.   0. ]
 [ 0.  12.8  0. ]
 [ 0.  13.6  0. ]
 [ 0.  14.4  0. ]
 [ 0.  15.2  0. ]
 [ 0.  16.   0. ]]

The following function creates a uniform beam from the previous parameters. On top of assigning the previous parameters, it defines other variables such as the connectivities between nodes

[13]:

wing.StructuralInformation.generate_uniform_beam(node_r,
                    mass_per_unit_length,
                    mass_iner_x,
                    mass_iner_y,
                    mass_iner_z,
                    pos_cg_B,
                    EA,
                    GAy,
                    GAz,
                    GJ,
                    EIy,
                    EIz,
                    num_node_elem = wing.StructuralInformation.num_node_elem,
                    y_BFoR = 'x_AFoR',
                    num_lumped_mass=0)

If we now show the connectivities between nodes, the function has created them for us. Further information

[14]:

print(wing.StructuralInformation.connectivities)

[[ 0  2  1]
 [ 2  4  3]
 [ 4  6  5]
 [ 6  8  7]
 [ 8 10  9]
 [10 12 11]
 [12 14 13]
 [14 16 15]
 [16 18 17]
 [18 20 19]]

Let’s define the boundary conditions as clamped for node 0 and free for the last node (-1) Further information

[15]:

wing.StructuralInformation.boundary_conditions[0] = 1
wing.StructuralInformation.boundary_conditions[-1] = -1

Aerodynamics¶

We need to define the number of panels in the wake (wake_panels) and the camber line of the wing which, in this case, is flat. Further information

[16]:

# Compute the number of panels in the wake (streamwise direction) based on the previous paramete
wake_panels = int(wake_length*chord/dt)

# Define the coordinates of the camber line of the wing
wing_camber = np.zeros((1, num_points_camber, 2))
wing_camber[0, :, 0] = np.linspace(0, 1, num_points_camber)

The following function creates an aerodynamic surface uniform on top of the beam that we have already created

[17]:

# Generate blade aerodynamics
wing.AerodynamicInformation.create_one_uniform_aerodynamics(wing.StructuralInformation,
                                 chord = chord,
                                 twist = 0.,
                                 sweep = 0.,
                                 num_chord_panels = num_chord_panels,
                                 m_distribution = 'uniform',
                                 elastic_axis = 0.5,
                                 num_points_camber = num_points_camber,
                                 airfoil = wing_camber)

Summary¶

Now, we have all the inputs that we need for SHARPy. In this section, a first simulation with SHARPy is run. However, it will perform no computation, it will just load the data so we can plot the system we have just created.

SHARPy runs a series of solvers and postprocessors in the order indicated by the flow variable in the SHARPy dictionary.

The generate cases module also allows us to show all the avaiable solvers and all the parameters they accept as inputs. To do so, we need to run the following commands:

[18]:

# Gather data about available solvers
SimInfo = gc.SimulationInformation() # Initialises the SimulationInformation class
SimInfo.set_default_values()         # Assigns the default values to all the solvers

# Print the available solvers and postprocessors
for key in SimInfo.solvers.keys():
    print(key)

_BaseStructural
AerogridLoader
BeamLoader
DynamicCoupled
DynamicUVLM
LinDynamicSim
LinearAssembler
Modal
NoAero
NonLinearDynamic
NonLinearDynamicCoupledStep
NonLinearDynamicMultibody
NonLinearDynamicPrescribedStep
NonLinearStatic
PrescribedUvlm
RigidDynamicCoupledStep
RigidDynamicPrescribedStep
StaticCoupled
StaticCoupledRBM
StaticTrim
StaticUvlm
StepLinearUVLM
StepUvlm
_BaseTimeIntegrator
NewmarkBeta
GeneralisedAlpha
Trim
LiftDistribution
Cleanup
AerogridPlot
BeamPlot
UDPout
PlotFlowField
PickleData
SaveParametricCase
FrequencyResponse
StallCheck
WriteVariablesTime
CreateSnapshot
SHARPy
SaveData
AeroForcesCalculator
AsymptoticStability
BeamLoads
FloatingForces
ShearVelocityField
DynamicControlSurface
GustVelocityField
TrajectoryGenerator
TurbVelocityField
SteadyVelocityField
ModifyStructure
StraightWake
HelicoidalWake
GridBox
BumpVelocityField
TurbVelocityFieldBts

Let’s output as an example, the input parameters of the BeamLoader solver. This solver is in charge of loading the structural infromation

[19]:

SimInfo.solvers['BeamLoader']

[19]:

{'unsteady': True, 'orientation': [1.0, 0, 0, 0], 'for_pos': [0.0, 0, 0]}

The following dictionary defines the basic inputs of SHARPy including the solvers to be run (flow variable), the case name and the route in the file system. In this case we are turning off the screen output because it is useless because we are not running any computation.

[20]:

SimInfo.solvers['SHARPy']['flow'] = ['BeamLoader',
                        'AerogridLoader']

SimInfo.solvers['SHARPy']['case'] = 'plot'
SimInfo.solvers['SHARPy']['route'] = './'
SimInfo.solvers['SHARPy']['write_screen'] = 'off'

We do not modify any of the default input paramteters in BeamLoader but we need to define the initial wake shape that we want and its size (wake_panels)

[21]:

SimInfo.solvers['AerogridLoader']['unsteady'] = 'on'
SimInfo.solvers['AerogridLoader']['mstar'] = wake_panels
SimInfo.solvers['AerogridLoader']['freestream_dir'] = np.array([1.,0.,0.])
SimInfo.solvers['AerogridLoader']['wake_shape_generator'] = 'StraightWake'
SimInfo.solvers['AerogridLoader']['wake_shape_generator_input'] = {'u_inf': WSP,
                                                                   'u_inf_direction' : np.array(
                                                                                         [np.cos(aoa_ini_deg*deg2rad),
                                                                                         0.,
                                                                                         np.sin(aoa_ini_deg*deg2rad)]),
                                                                   'dt': dt}

The following functions write the input files needed by SHARPy

[22]:

gc.clean_test_files(SimInfo.solvers['SHARPy']['route'], SimInfo.solvers['SHARPy']['case'])
wing.generate_h5_files(SimInfo.solvers['SHARPy']['route'], SimInfo.solvers['SHARPy']['case'])
SimInfo.generate_solver_file()

The following line of code runs SHARPy inside the jupyter notebook. It is equivalent to run in a terminal: sharpy plot.sharpy being “plot” the case name defined above

[23]:

sharpy_output = sharpy.sharpy_main.main(['',
                                         SimInfo.solvers['SHARPy']['route'] +
                                         SimInfo.solvers['SHARPy']['case'] +
                                         '.sharpy'])

Let’s plot the case we have created. The function below provides a quick way of generating an interactive plot but it is not suitable for large cases. Luckily SHARPy has other more complex plotting capabilities.

If you download the original jupyter notebook, change thee geometric parameters above and rerun the jupyter notebook you will see its impact on the geometry.

Only the first 6 wake panels are plotted for efficiency.

[24]:

pu.plot_timestep(sharpy_output, tstep=-1, minus_mstar=(wake_panels - 6), plotly=True)

Static simulation¶

Next, we run a static simulation of the previous system. We have already defined the required inputs for BeamLoader and AerogridLoader but we need to define the static solvers. We are going to use a FSI solver (StaticCoupled) that will include a structural solver (NonLinearStatic) and an aerodynamic solver (StaticUvlm).

We use most of the default values but we redefine some of them: - We assign the air density to all the solvers that need it - We turn off the gravity on the structural solver - We use a horseshoe solution without roll up for the aerodynamics - We define the velocity field against the wing - We do not use load steps in the FSI solver

In the documentation you can fine further information about the modular framework and about the solvers and their inputs.

[25]:

# Define the simulation
SimInfo.solvers['SHARPy']['flow'] = ['BeamLoader',
                        'AerogridLoader',
                        'StaticCoupled']

SimInfo.set_variable_all_dicts('rho', air_density)

SimInfo.solvers['SHARPy']['case'] = 'static'
SimInfo.solvers['SHARPy']['write_screen'] = 'on'

SimInfo.solvers['NonLinearStatic']['gravity_on'] = False

SimInfo.solvers['StaticUvlm']['horseshoe'] = True
SimInfo.solvers['StaticUvlm']['n_rollup'] = 0
SimInfo.solvers['StaticUvlm']['velocity_field_generator'] = 'SteadyVelocityField'
SimInfo.solvers['StaticUvlm']['velocity_field_input'] = {'u_inf' : WSP,
                                                         'u_inf_direction' : np.array(
                                                                                [np.cos(aoa_ini_deg*deg2rad),
                                                                                 0.,
                                                                                 np.sin(aoa_ini_deg*deg2rad)])}

SimInfo.solvers['StaticCoupled']['structural_solver'] = 'NonLinearStatic'
SimInfo.solvers['StaticCoupled']['structural_solver_settings'] = SimInfo.solvers['NonLinearStatic']
SimInfo.solvers['StaticCoupled']['aero_solver'] = 'StaticUvlm'
SimInfo.solvers['StaticCoupled']['aero_solver_settings'] = SimInfo.solvers['StaticUvlm']
SimInfo.solvers['StaticCoupled']['n_load_steps'] = 0

The following functions create the input files required by SHARPy. For further information check: - Configuration file - FEM input file - Aerodynamic input file

[26]:

gc.clean_test_files(SimInfo.solvers['SHARPy']['route'], SimInfo.solvers['SHARPy']['case'])
SimInfo.generate_solver_file()
wing.generate_h5_files(SimInfo.solvers['SHARPy']['route'], SimInfo.solvers['SHARPy']['case'])

[27]:

# Running SHARPy again inside jupyter
sharpy_output = sharpy.sharpy_main.main(['',
                                         SimInfo.solvers['SHARPy']['route'] +
                                         SimInfo.solvers['SHARPy']['case'] +
                                         '.sharpy'])

--------------------------------------------------------------------------------
            ######  ##     ##    ###    ########  ########  ##    ##
           ##    ## ##     ##   ## ##   ##     ## ##     ##  ##  ##
           ##       ##     ##  ##   ##  ##     ## ##     ##   ####
            ######  ######### ##     ## ########  ########     ##
                 ## ##     ## ######### ##   ##   ##           ##
           ##    ## ##     ## ##     ## ##    ##  ##           ##
            ######  ##     ## ##     ## ##     ## ##           ##
--------------------------------------------------------------------------------
Aeroelastics Lab, Aeronautics Department.
    Copyright (c), Imperial College London.
    All rights reserved.
    License available at https://github.com/imperialcollegelondon/sharpy
Running SHARPy from /home/arturo/code/sharpy/docs/source/content/example_notebooks
SHARPy being run is in /home/arturo/code/sharpy
The branch being run is dev_setting_error
The version and commit hash are: v1.2.1-360-gc3031060-c3031060
SHARPy output folder set
     ./output//static/
Generating an instance of BeamLoader
Generating an instance of AerogridLoader
Variable dx1 has no assigned value in the settings file.
    will default to the value: -1.0
Variable ndx1 has no assigned value in the settings file.
    will default to the value: 1
Variable r has no assigned value in the settings file.
    will default to the value: 1.0
Variable dxmax has no assigned value in the settings file.
    will default to the value: -1.0
The aerodynamic grid contains 1 surfaces
  Surface 0, M=4, N=20
     Wake 0, M=400, N=20
  In total: 80 bound panels
  In total: 8000 wake panels
  Total number of panels = 8080
Generating an instance of StaticCoupled
Generating an instance of NonLinearStatic
Generating an instance of StaticUvlm



|=====|=====|============|==========|==========|==========|==========|==========|==========|
|iter |step | log10(res) |    Fx    |    Fy    |    Fz    |    Mx    |    My    |    Mz    |
|=====|=====|============|==========|==========|==========|==========|==========|==========|
|  0  |  0  |  0.00000   | -0.0187  | -0.0006  |  0.6029  |  4.8229  |  0.1523  |  0.1495  |
|  1  |  0  |  -9.22874  | -0.0188  | -0.0006  |  0.6043  |  4.8366  |  0.1526  |  0.1504  |
FINISHED - Elapsed time = 1.1078096 seconds
FINISHED - CPU process time = 4.8641626 seconds

The next function will plot the static equilibirium solution for the wing

[28]:

pu.plot_timestep(sharpy_output, tstep=-1, minus_mstar=(wake_panels-6), plotly=True)

Dynamic simulation¶

Finally, we run a dynamic simulation after the previous static one. With that objective, we include a dynamic FSI solver (DynamicCoupled) in the simulation

[29]:

SimInfo.solvers['SHARPy']['flow'] = ['BeamLoader',
                        'AerogridLoader',
                        'StaticCoupled',
                        'DynamicCoupled']

SimInfo.solvers['SHARPy']['route'] = './'
SimInfo.solvers['SHARPy']['case'] = 'dynamic'

We need information about the time step and the number of iterations

[30]:

# Compute the number of time steps needed based on the previous parameters
time_steps = int(end_time/dt)

# Define the time step and the number of time steps in every solver that requires them as input
SimInfo.set_variable_all_dicts('dt', dt)
SimInfo.define_num_steps(time_steps)

We define the approximation we are going to use for the wake convection (convection_scheme), the velocity field and deactivate the gravity.

We are going to use a FSI solver (DynamicCoupled) that couples a structual solver (NonLinearDynamicPrescribedStep) and an aerodynamic solver (StepUvlm).

Moreover, the FSI solver runs a series of postprocessors after each time step. Specifically it will save a plot of the structure (BeamPlot) and of the aerodynamic surfaces (AerogridPlot).

[31]:

SimInfo.solvers['StepUvlm']['convection_scheme'] = 2
SimInfo.solvers['StaticUvlm']['velocity_field_generator'] = 'SteadyVelocityField'
SimInfo.solvers['StepUvlm']['velocity_field_input'] = {'u_inf' : WSP,
                                                       'u_inf_direction' : np.array(
                                                                              [np.cos(aoa_end_deg*deg2rad),
                                                                               0.,
                                                                               np.sin(aoa_end_deg*deg2rad)])}

SimInfo.solvers['NonLinearDynamicPrescribedStep']['gravity_on'] = False

SimInfo.solvers['DynamicCoupled']['structural_solver'] = 'NonLinearDynamicPrescribedStep'
SimInfo.solvers['DynamicCoupled']['structural_solver_settings'] = SimInfo.solvers['NonLinearDynamicPrescribedStep']
SimInfo.solvers['DynamicCoupled']['aero_solver'] = 'StepUvlm'
SimInfo.solvers['DynamicCoupled']['aero_solver_settings'] = SimInfo.solvers['StepUvlm']
SimInfo.solvers['DynamicCoupled']['postprocessors'] = ['BeamPlot', 'AerogridPlot']
SimInfo.solvers['DynamicCoupled']['postprocessors_settings'] = {'BeamPlot': SimInfo.solvers['BeamPlot'],
                                                             'AerogridPlot': SimInfo.solvers['AerogridPlot']}

Finally we need to define the movement of the whole wing through its velocity and acceleration. In this case we want a clamped structure so we will set every thing to zero.

Moreover, se can define forces applied togheter with the aerodynamic ones. In this case they will be set to zero.

[32]:

SimInfo.with_forced_vel = True
SimInfo.for_vel = np.zeros((time_steps,6), dtype=float)
SimInfo.for_acc = np.zeros((time_steps,6), dtype=float)
SimInfo.with_dynamic_forces = True
SimInfo.dynamic_forces = np.zeros((time_steps,wing.StructuralInformation.num_node,6),
                                  dtype=float)

We write the input files required by SHARPy again:

[33]:

gc.clean_test_files(SimInfo.solvers['SHARPy']['route'], SimInfo.solvers['SHARPy']['case'])
wing.generate_h5_files(SimInfo.solvers['SHARPy']['route'], SimInfo.solvers['SHARPy']['case'])
SimInfo.generate_solver_file()
SimInfo.generate_dyn_file(time_steps)

And we run the simulation

[34]:

sharpy_output = sharpy.sharpy_main.main(['',
                                         SimInfo.solvers['SHARPy']['route'] +
                                         SimInfo.solvers['SHARPy']['case'] +
                                         '.sharpy'])

--------------------------------------------------------------------------------
            ######  ##     ##    ###    ########  ########  ##    ##
           ##    ## ##     ##   ## ##   ##     ## ##     ##  ##  ##
           ##       ##     ##  ##   ##  ##     ## ##     ##   ####
            ######  ######### ##     ## ########  ########     ##
                 ## ##     ## ######### ##   ##   ##           ##
           ##    ## ##     ## ##     ## ##    ##  ##           ##
            ######  ##     ## ##     ## ##     ## ##           ##
--------------------------------------------------------------------------------
Aeroelastics Lab, Aeronautics Department.
    Copyright (c), Imperial College London.
    All rights reserved.
    License available at https://github.com/imperialcollegelondon/sharpy
Running SHARPy from /home/arturo/code/sharpy/docs/source/content/example_notebooks
SHARPy being run is in /home/arturo/code/sharpy
The branch being run is dev_setting_error
The version and commit hash are: v1.2.1-360-gc3031060-c3031060
SHARPy output folder set
     ./output//dynamic/
Generating an instance of BeamLoader
Generating an instance of AerogridLoader
Variable dx1 has no assigned value in the settings file.
    will default to the value: -1.0
Variable ndx1 has no assigned value in the settings file.
    will default to the value: 1
Variable r has no assigned value in the settings file.
    will default to the value: 1.0
Variable dxmax has no assigned value in the settings file.
    will default to the value: -1.0
The aerodynamic grid contains 1 surfaces
  Surface 0, M=4, N=20
     Wake 0, M=400, N=20
  In total: 80 bound panels
  In total: 8000 wake panels
  Total number of panels = 8080
Generating an instance of StaticCoupled
Generating an instance of NonLinearStatic
Generating an instance of StaticUvlm



|=====|=====|============|==========|==========|==========|==========|==========|==========|
|iter |step | log10(res) |    Fx    |    Fy    |    Fz    |    Mx    |    My    |    Mz    |
|=====|=====|============|==========|==========|==========|==========|==========|==========|
|  0  |  0  |  0.00000   | -0.0187  | -0.0006  |  0.6029  |  4.8229  |  0.1523  |  0.1495  |
|  1  |  0  |  -9.22874  | -0.0188  | -0.0006  |  0.6043  |  4.8366  |  0.1526  |  0.1504  |
Generating an instance of DynamicCoupled
Generating an instance of NonLinearDynamicPrescribedStep
Generating an instance of StepUvlm
Generating an instance of BeamPlot
Generating an instance of AerogridPlot



|=======|========|======|==============|==============|==============|==============|==============|
|  ts   |   t    | iter | struc ratio  |  iter time   | residual vel |  FoR_vel(x)  |  FoR_vel(z)  |
|=======|========|======|==============|==============|==============|==============|==============|
|   1   | 0.1250 |  4   |   0.422637   |   2.156250   |  -5.501499   | 0.000000e+00 | 0.000000e+00 |
|   2   | 0.2500 |  3   |   0.536941   |   1.912537   |  -5.442787   | 0.000000e+00 | 0.000000e+00 |
|   3   | 0.3750 |  3   |   0.524459   |   2.198519   |  -5.640521   | 0.000000e+00 | 0.000000e+00 |
|   4   | 0.5000 |  3   |   0.312450   |   1.014755   |  -5.645651   | 0.000000e+00 | 0.000000e+00 |
|   5   | 0.6250 |  3   |   0.516388   |   1.390178   |  -5.889684   | 0.000000e+00 | 0.000000e+00 |
|   6   | 0.7500 |  3   |   0.493723   |   1.473236   |  -5.914635   | 0.000000e+00 | 0.000000e+00 |
|   7   | 0.8750 |  3   |   0.364603   |   1.169983   |  -6.040009   | 0.000000e+00 | 0.000000e+00 |
|   8   | 1.0000 |  3   |   0.375979   |   1.130078   |  -5.925998   | 0.000000e+00 | 0.000000e+00 |
|   9   | 1.1250 |  3   |   0.422528   |   1.220274   |  -5.836958   | 0.000000e+00 | 0.000000e+00 |
|  10   | 1.2500 |  3   |   0.470494   |   1.508245   |  -5.632144   | 0.000000e+00 | 0.000000e+00 |
|  11   | 1.3750 |  3   |   0.518558   |   1.643736   |  -5.513670   | 0.000000e+00 | 0.000000e+00 |
|  12   | 1.5000 |  3   |   0.461596   |   1.443260   |  -5.471589   | 0.000000e+00 | 0.000000e+00 |
|  13   | 1.6250 |  3   |   0.382144   |   1.202683   |  -5.423647   | 0.000000e+00 | 0.000000e+00 |
|  14   | 1.7500 |  3   |   0.381228   |   1.303255   |  -5.451292   | 0.000000e+00 | 0.000000e+00 |
|  15   | 1.8750 |  3   |   0.488247   |   1.640878   |  -5.503924   | 0.000000e+00 | 0.000000e+00 |
|  16   | 2.0000 |  3   |   0.533105   |   2.012032   |  -5.521127   | 0.000000e+00 | 0.000000e+00 |
|  17   | 2.1250 |  3   |   0.488828   |   1.673044   |  -5.649248   | 0.000000e+00 | 0.000000e+00 |
|  18   | 2.2500 |  3   |   0.408699   |   1.435696   |  -5.852560   | 0.000000e+00 | 0.000000e+00 |
|  19   | 2.3750 |  3   |   0.549950   |   1.755136   |  -6.169248   | 0.000000e+00 | 0.000000e+00 |
|  20   | 2.5000 |  3   |   0.496495   |   1.671328   |  -6.501434   | 0.000000e+00 | 0.000000e+00 |
|  21   | 2.6250 |  3   |   0.388377   |   1.389508   |  -5.911316   | 0.000000e+00 | 0.000000e+00 |
|  22   | 2.7500 |  3   |   0.579859   |   2.018747   |  -5.844249   | 0.000000e+00 | 0.000000e+00 |
|  23   | 2.8750 |  3   |   0.418595   |   1.542469   |  -5.662014   | 0.000000e+00 | 0.000000e+00 |
|  24   | 3.0000 |  3   |   0.478950   |   1.563416   |  -5.623002   | 0.000000e+00 | 0.000000e+00 |
|  25   | 3.1250 |  3   |   0.414521   |   1.306833   |  -5.695734   | 0.000000e+00 | 0.000000e+00 |
|  26   | 3.2500 |  3   |   0.444346   |   1.431409   |  -5.679340   | 0.000000e+00 | 0.000000e+00 |
|  27   | 3.3750 |  3   |   0.468878   |   1.387275   |  -5.742054   | 0.000000e+00 | 0.000000e+00 |
|  28   | 3.5000 |  3   |   0.349114   |   1.171457   |  -5.934912   | 0.000000e+00 | 0.000000e+00 |
|  29   | 3.6250 |  3   |   0.347932   |   1.166252   |  -6.170404   | 0.000000e+00 | 0.000000e+00 |
|  30   | 3.7500 |  3   |   0.413979   |   1.302708   |  -6.498959   | 0.000000e+00 | 0.000000e+00 |
|  31   | 3.8750 |  3   |   0.365911   |   1.211784   |  -6.054012   | 0.000000e+00 | 0.000000e+00 |
|  32   | 4.0000 |  3   |   0.445476   |   1.360627   |  -6.023555   | 0.000000e+00 | 0.000000e+00 |
|  33   | 4.1250 |  3   |   0.413054   |   1.284355   |  -5.782342   | 0.000000e+00 | 0.000000e+00 |
|  34   | 4.2500 |  3   |   0.467896   |   1.423890   |  -5.673256   | 0.000000e+00 | 0.000000e+00 |
|  35   | 4.3750 |  3   |   0.380931   |   1.219211   |  -5.733463   | 0.000000e+00 | 0.000000e+00 |
|  36   | 4.5000 |  3   |   0.512592   |   1.547063   |  -5.705716   | 0.000000e+00 | 0.000000e+00 |
|  37   | 4.6250 |  3   |   0.444554   |   1.350007   |  -5.751720   | 0.000000e+00 | 0.000000e+00 |
|  38   | 4.7500 |  3   |   0.449910   |   1.374741   |  -5.851236   | 0.000000e+00 | 0.000000e+00 |
|  39   | 4.8750 |  3   |   0.444920   |   1.371475   |  -5.920499   | 0.000000e+00 | 0.000000e+00 |
|  40   | 5.0000 |  3   |   0.418596   |   1.348310   |  -6.297188   | 0.000000e+00 | 0.000000e+00 |
...Finished
FINISHED - Elapsed time = 69.6649866 seconds
FINISHED - CPU process time = 323.5800926 seconds

This is the plot of the wing at the end of the simulation

[35]:

pu.plot_timestep(sharpy_output, tstep=-1, minus_mstar=(wake_panels-6), plotly=True)

Postprocessing¶

As an example, we are going to plot the evolution of the tip position along time

[36]:

time = np.linspace(0, dt*time_steps, time_steps)
tip_pos = np.zeros((time_steps))
for it in range(time_steps):
    tip_pos[it] = sharpy_output.structure.timestep_info[it].pos[-1, 2]

fig, plots = plt.subplots(1, 1, figsize=(6, 3))

plots.grid()
plots.set_xlabel("time [s]")
plots.set_ylabel("tip position [m]")
plots.plot(time, tip_pos, '-')

plt.show()

_images/content_example_notebooks_cantilever_wing_66_0.svg

Conclusions¶

This notebook provides the big picture of the simulations with SHARPy. We recommend to go through the documentation with special attention to the SHARPy file description and the examples.

Downloadable files¶

Input data for wind turbine:

./example_notebooks/source/type02_db_NREL5MW_v02.xlsx

Input data for static cantilever:

./example_notebooks/cantilever/model_static_cantilever.py

Note: some of these examples may need additional files which would be located in the ./example_notebooks/ directory. It is recommended that you run these cases directly from there rather than downloading. If you download, make sure you replicate the folder structure such that the examples are capable of finding the required files.

Contributing to SHARPy¶

Bug fixes and features¶

SHARPy is a collaborative effort, and this means that some coding practices need to be encouraged so that the code is kept tidy and consistent. Any user is welcome to raise issues for bug fixes and feature proposals through Github.

If you are submitting a bug report:

Make sure your SHARPy, xbeam and uvlm local copies are up to date and in the same branch.
Double check that your python distribution is updated by comparing with the utils/environment_*.yml file.
Try to assemble a minimal working example that can be run quickly and easily.
Describe as accurately as possible your setup (OS, path, compilers…) and the problem.
Raise an issue with all this information in the Github repo and label it potential bug.

Please bear in mind that we do not have the resources to provide support for user modifications of the code through Github. If you have doubts about how to modify certain parts of the code, contact us through email and we will help you as much as we can.

If you are fixing a bug:

THANKS!
Please create a pull request from your modified fork, and describe in a few lines which bug you are fixing, a minimal example that triggers the bug and how you are fixing it. We will review it ASAP and hopefully it will be incorporated in the code!

If you have an idea for new functionality but do not know how to implement it:

We welcome tips and suggestions from users, as it allow us to broaden the scope of the code. The more people using it, the better!
Feel free to fill an issue in Github, and tag it as feature proposal. Please understand that the more complete the description of the potential feature, the more likely it is that some of the developers will give it a go.

If you have developed new functionality and you want to share it with the world:

AWESOME! Please follow the same instructions than for the bug fix submission. If you have some peer-reviewed references related to the new code, even better, as it will save us some precious time.

Code formatting¶

We try to follow the PEP8 standards (with spaces, no tabs please!) and Google Python Style Guide. We do not ask you to freak out over formatting, but please, try to keep it tidy and descriptive. A good tip is to run pylint https://www.pylint.org/ to make sure there are no obvious formatting problems.

Documentation¶

Contributing to SHARPy’s documentation benefits everyone. As a developer, writing documentation helps you better understand what you have done and whether your functions etc make logical sense. As a user, any documentation is better than digging through the code. The more we have documented, the easier the code is to use and the more users we can have.

If you want to contribute by documenting code, you have come to the right place.

SHARPy is documented using Sphinx and it extracts the documentation directly from the source code. It is then sorted into directories automatically and a human readable website generated. The amount of work you need to do is minimal. That said, the recipe for a successfully documented class, function, module is the following:

Your documentation has to be written in ReStructuredText (rst). I know, another language… hence I will leave a few tips:
- Inline code is written using two backticks ``
- Inline math is written as :math:`1+\exp^{i\pi} = 0`. Don’t forget the backticks!
- Math in a single or multiple lines is simple:
```
    .. math:: 1 + \exp{i\pi} = 0
```
- Lists in ReStructuredText are tricky, I must admit. Therefore, I will link to some examples. The key resides in not forgetting the spaces, in particular when you go onto a second line!
- The definite example list can be found here.
Titles and docstrings, the bare minimum:
- Start docstrings with r such that they are interpreted raw:
```
r"""
My docstring
"""
```
- All functions, modules and classes should be given a title that goes in the first line of the docstring
- If you are writing a whole package with an __init__.py file, even if it’s empty, give it a human readable docstring. This will then be imported into the documentation
- For modules with several functions, the module docstring has to be at the very top of the file, prior to the import statements.
We use the Google documentation style. A very good set of examples of Google style documentation for functions, modules, classes etc. can be found here.

Function arguments and returns:

Function arguments are simple to describe:

def func(arg1, arg2):
"""Summary line.

Extended description of function.

Args:
  arg1 (int): Description of arg1
  arg2 (str): Description of arg2

Returns:
  bool: Description of return value

"""
    return True

Solver settings:
- If your code has a settings dictionary, with defaults and types then make sure that:
  - They are defined as class variables and not instance attributes.
  - Define a settings_types, settings_default and settings_description dictionaries.
  - After all your settings, update the docstring with the automatically generated settings table. You will need to import the sharpy.utils.settings module
    settings_types = dict() settings_default = dict() settings_description = dict() # keep adding settings settings_table = sharpy.utils.settings.SettingsTable() __doc__ += settings_table.generate(settings_types, settings_default ,settings_description)
See how your docs looks like!
- Once you are done, run the following SHARPy command:
```
sharpy any_string -d
```
- If you are making minor updates to docstrings (i.e. you are not documenting a previously undocumented function/class/module) you can simply change directory to sharpy/docs and run
```
make html
```
- Your documentation will compile and warnings will appear etc. You can check the result by opening
```
docs/build/index.html
```
and navigating to your recently created page.
- Make sure that before committing any changes in the documentation you update the entire docs directory by running
```
sharpy any_string -d
```

Thank you for reading through this and contributing to make SHARPy a better documented, more user friendly code!

Git branching model¶

For the development of SHARPy, we try to follow this branching model summarised by the schematic

BranchingModel Credit: Vincent Driessen https://nvie.com/posts/a-successful-git-branching-model/

Therefore, attending to this model our branches have the following versions of the code:

master: latest stable release - paired with the appropriate tag.
develop: latest stable development build. Features get merged to develop.
rc-**: release candidate branch. Prior to releasing tests are performed on this branch.
dev_doc: documentation development branch. All work relating to documentation gets done here.
fix_**: hotfix branch.
dev_**: feature development branch.

If you contribute, please make sure you know what branch to work from. If in doubt please ask!

Commit names are also important since they are the backbone of the code’s change log. Please write concise commit titles and explain the main changes in the body of the commit message. An excellent guide on writing good commit messages can be found here.

For developers:¶

Releasing a new version¶

In the release candidate branch:

Update the version number in the docs configuration file docs/conf.py
Update version.json file
Update version in sharpy/__init__.py file
Commit, push and wait for tests to pass
Merge release candidate branch into master branch

In the master branch:

Run the github_changelog_generator tool locally with the following parameters:

github_changelog_generator -u imperialcollegelondon -p sharpy -t <your_github_token> --future-release <new_release_version>

Create a release tag. IMPORTANT: ensure it is an annotated tag, otherwise the version and commit number in SHARPy will not display properly
```
git tag -a <tagname>
git push origin --tags -f
```
where <tagname> is something like 2.0.
Create the GitHub release, choosing the newly created tag from the dropdown menu. Do not create a tag from the dropdown menu directly because it will not be an annotated tag

The SHARPy Case files¶

SHARPy takes as input a series of .h5 files that contain the numerical data and a .sharpy file that contains the settings for each of the solvers. How these files are generated is at the user’s discretion, though templates are provided, and all methods are valid as long as the required variables are provided with the appropriate format.

Modular Framework¶

SHARPy is built with a modular framework in mind. The following diagram shows the strutuctre of a nonlinear, time marching aeroelastic simulation

Each of the blocks correspond to individual solvers with specific settings. How we choose which solvers to run, in which order and with what settings is done through the solver configuration file, explained in the next section.

Solver configuration file¶

The solver configuration file is the main input to SHARPy. It is a ConfigObj formatted file with the .sharpy extension. It contains the settings for each of the solvers and the order in which to run them.

A typical way to assemble the solver configuration file is to place all your desired settings in a dictionary and then convert to and write your ConfigObj. If a setting is not provided the default value will be used. The settings that each solver takes, its type and default value are explained in their relevant documentation pages.

import configobj
filename = '<case_route>/<case_name>.sharpy'
config = configobj.ConfigObj()
config.filename = filename
config['SHARPy'] = {'case': '<your SHARPy case name>',  # an example setting
                    # Rest of your settings for the PreSHARPy class
                    }
config['BeamLoader'] = {'orientation': [1., 0., 0.],  # an example setting
                        # Rest of settings for the BeamLoader solver
                        }
# Continue as above for the remainder of solvers that you would like to include

# finally, write the config file
config.write()

The resulting .sharpy file is a plain text file with your specified settings for each of the solvers.

Note that, therefore, if one of your settings is a np.array, it will get transformed into a string of plain text before being read by SHARPy. However, any setting with list(float) specified as its setting type will get converted into a np.array once it is read by SHARPy.

FEM file¶

The case.fem.h5 file has several components. We go one by one:

num_node_elem [int] : number of nodes per element.

Always 3 in our case (3 nodes per structural elements - quadratic beam elements).
num_elem [int] : number of structural elements.
num_node [int] : number of nodes.

For simple structures, it is num_elem*(num_node_elem - 1) - 1. For more complicated ones, you need to calculate it properly.
coordinates [num_node, 3]: coordinates of the nodes in body-attached FoR (A).
connectivites [num_elem, num_node_elem] : Beam element’s connectivities.

Every row refers to an element, and the three integers in that row are the indices of the three nodes belonging to that elem. Now, the catch: the ordering is not as you’d think. Order them as [0, 2, 1]. That means, first one, last one, central one. The following image shows the node indices inside the circles representing the nodes, the element indices in blue and the resulting connectivities matrix next to it. Connectivities are tricky when considering complex configurations. Pay attention at the beginning and you’ll save yourself a lot of trouble.
stiffness_db [:, 6, 6]: database of stiffness matrices.

The first dimension has as many elements as different stiffness matrices are in the model.
elem_stiffness [num_elem] : array of indices (starting at 0).

It links every element (index) to the stiffness matrix index in stiffness_db. For example elem_stiffness[0] = 0 ; elem_stiffness[2] = 1 means that the element 0 has a stiffness matrix equal to stiffness_db[0, :, :] , and the second element has a stiffness matrix equal to stiffness_db[1, :, :].

The shape of a stiffness matrix, $\mathrm{S}$ is:

\[\begin{split}\mathrm{S} = \begin{bmatrix} EA & & & & & \\ & GA_y & & & & \\ & & GA_z & & & \\ & & & GJ & & \\ & & & & EI_y & \\ & & & & & EI_z \\ \end{bmatrix}\end{split}\]

with the cross terms added if needed.

mass_db and elem_mass follow the same scheme than the stiffness, but the mass matrix is given by:

\[\begin{split}\mathrm{M} = \begin{bmatrix} m\mathbf{I} & -\tilde{\boldsymbol{\xi}}_{cg}m \\ \tilde{\boldsymbol{\xi}}_{cg}m & \mathbf{J}\\ \end{bmatrix}\end{split}\]

where $m$ is the distributed mass per unit length $kg/m$ , $(\tilde{\bullet})$ is the skew-symmetric matrix of a vector and $\boldsymbol{\xi}_{cg}$ is the location of the centre of gravity with respect to the elastic axis in MATERIAL (local) FoR. And what is the Material FoR? This is an important point, because all the inputs that move WITH the beam are in material FoR. For example: follower forces, stiffness, mass, lumped masses…

The material frame of reference is noted as $B$. Essentially, the $x$ component is tangent to the beam in the increasing node ordering, $z$ looks up generally and $y$ is oriented such that the FoR is right handed.

In the practice (vertical surfaces, structural twist effects…) it is more complicated than this. The only sure thing about $B$ is that its $x$ direction is tangent to the beam in the increasing node number direction. However, with just this, we have an infinite number of potential reference frames, with $y$ and $z$ being normal to $x$ but rotating around it. The solution is to indicate a for_delta, or frame of reference delta vector ($\Delta$).

Now we can define unequivocally the material frame of reference. With $x_B$ and $\Delta$ defining a plane, $y_b$ is chosen such that the $z$ component is oriented upwards with respect to the lifting surface.

From this definition comes the only constraint to $\Delta$: it cannot be parallel to $x_B$.
frame_of_reference_delta [num_elem, num_node_elem, 3]: rotation vector to FoR $B$.

contains the $\Delta$ vector in body-attached ($A$) frame of reference.

As a rule of thumb:

\[\begin{split}\Delta = \begin{cases} [-1, 0, 0], \quad \text{if right wing} \\ [1, 0, 0], \quad \text{if left wing} \\ [0, 1, 0], \quad \text{if fuselage} \\ [-1, 0, 0], \quad \text{if vertical fin} \\ \end{cases}\end{split}\]

These rules of thumb only work if the nodes increase towards the tip of the surfaces (and the tail in the case of the fuselage).
structural_twist [num_elem, num_node_elem]: Element twist.

Technically not necessary, as the same effect can be achieved with FoR_delta.
boundary_conditions [num_node]: boundary conditions.
An array of integers (np.zeros((num_node, ), dtype=int)) and contains all 0 except for
One node NEEDS to have a 1 , this is the reference node. Usually, the first node has 1 and is located in [0, 0, 0]. This makes things much easier.

If the node is a tip of a beam (is not attached to 2 elements, but just 1), it needs to have a -1.
beam_number [num_elem]: beam index.

Is another array of integers. Usually you don’t need to modify its value. Leave it at 0.
app_forces [num_elem, 6]: applied forces and moments.

Contains the applied forces app_forces[:, 0:3] and moments app_forces[:, 3:6] in a given node.

Important points: the forces are given in Material FoR (check above). That means that in a symmetrical model, a thrust force oriented upstream would have the shape [0, T, 0, 0, 0, 0] in the right wing, while the left would be [0, -T, 0, 0, 0, 0]. Likewise, a torsional moment for twisting the wing leading edge up would be [0, 0, 0, M, 0, 0] for the right, and [0, 0, 0, -M, 0, 0] for the left. But careful, because an out-of-plane bending moment (wing tip up) has the same sign (think about it).
lumped_mass [:]: lumped masses.

Is an array with as many masses as needed (in kg this time). Their order is important, as more information is required to implement them in a model.
lumped_mass_nodes [:]: Lumped mass nodes.

Is an array of integers. It contains the index of the nodes related to the masses given in lumped_mass in order.
lumped_mass_inertia [:, 3, 3]: Lumped mass inertia.

Is an array of 3x3 inertial tensors. The relationship is set by the ordering as well.
lumped_mass_position [:, 3]: Lumped mass position.

Is the relative position of the lumped mass with respect to the node (given in lumped_masss_nodes ) coordinates. ATTENTION: the lumped mass is solidly attached to the node, and thus, its position is given in Material FoR.

Aerodynamics file¶

All the aerodynamic data is contained in case.aero.h5.

It is important to know that the input for aero is usually based on elements (and inside the elements, their nodes). This causes sometimes an overlap in information, as some nodes are shared by two adjacent elements (like in the connectivities graph in the previous section). The easier way of dealing with this is to make sure the data is consistent, so that the properties of the last node of the first element are the same than the first node of the second element.

Item by item:

airfoils: Airfoil group.

In the aero.h5 file, there is a Group called airfoils. The airfoils are stored in this group (which acts as a folder) as a two-column matrix with $x/c$ and $y/c$ in each column. They are named '0', '1' , and so on.
chords [num_elem, num_node_elem]: Chord

Is an array with the chords of every airfoil given in an element/node basis.
twist [num_elem, num_node_elem]: Twist.

Has the twist angle in radians. It is implemented as a rotation around the local $x$ axis.
sweep [num_elem, num_node_elem]: Sweep.

Same here, just a rotation around $z$.
airfoil_distribution [num_elem, num_node_elem]: Airfoil distribution.

Contains the indices of the airfoils that you put previously in airfoils.
surface_distribution [num_elem]: Surface integer array.

It contains the index of the surface the element belongs to. Surfaces need to be continuous, so please note that if your beam numbering is not continuous, you need to make a surface per continuous section.
surface_m [num_surfaces]: Chordwise panelling.

Is an integer array with the number of chordwise panels for every surface.
m_distribution [string]: Discretisation method.

Is a string with the chordwise panel distribution. In almost all cases, leave it at uniform.
aero_node [num_node]: Aerodynamic node definition.

Is a boolean (True or False) array that indicates if that node has a lifting surface attached to it.
elastic_axis [num_elem, num_node_elem]: elastic axis.

Indicates the elastic axis location with respect to the leading edge as a fraction of the chord of that rib. Note that the elastic axis is already determined, as the beam is fixed now, so this settings controls the location of the lifting surface wrt the beam.
control_surface [num_elem, num_node_elem]: Control surface.

Is an integer array containing -1 if that section has no control surface associated to it, and 0, 1, 2 ... if the section belongs to the control surface 0, 1, 2 ... respectively.
control_surface_type [num_control_surface]: Control Surface type.

Contains 0 if the control surface deflection is static, and 1 is it is dynamic.
control_surface_chord [num_control_surface]: Control surface chord.

Is an INTEGER array with the number of panels belonging to the control surface. For example, if M = 4 and you want your control surface to be $0.25c$, you need to put 1.
control_surface_hinge_coord [num_control_surface]: Control surface hinge coordinate.

Only necessary for lifting surfaces that are deflected as a whole, like some horizontal tails in some aircraft. Leave it at 0 if you are not modelling this.
airfoil_efficiency [num_elem, num_node_elem, 2, 3]: Airfoil efficiency.

This is an optional setting that introduces a user-defined efficiency and constant terms to the mapping between the aerodynamic forces calculated at the lattice grid and the structural nodes. The formatting of the 4-dimensional array is simple. The first two dimensions correspond to the element index and the local node index. The third index is whether the term is the multiplier to the force 0 or a constant term 1. The final term refers to, in the local, body-attached B frame, the factors and constant terms for: fy, fz, mx. For more information on how these factors are included in the mapping terms see sharpy.aero.utils.mapping.aero2struct_force_mapping().
polars Group (optional): Use airfoil polars to correct aerodynamic forces.

This is an optional group to add if correcting the aerodynamic forces using airfoil polars is desired. A polar should be included for each airfoil defined. Each entry consists of a 4-column table. The first column corresponds to the angle of attack (in radians) and then the C_L, C_D and C_M.

Time-varying force input file (`.dyn.h5`)¶

The .dyn.h5 file is an optional input file that may contain force and acceleration inputs that vary with time. This is intended for use in dynamic problems. For SHARPy to look for and use this file the setting unsteady in the BeamLoader must be turned to on.

Appropriate data entries in the .dyn.h5 include:

dynamic_forces [num_t_steps, num_node, 6]: Dynamic forces in body attached B frame.

Forces given at each time step, for each node and then for the 6 degrees of freedom (fx, fy, fz, mx, my, mz) in a body-attached (local) frame of reference B.
for_pos [num_t_steps, 6]: Body frame of reference (A FoR) position.

Position of the reference frame A in time.
for_vel [num_t_steps, 6]: Body frame of reference (A FoR) velocity.

Velocity of the reference frame A in time.
for_acc [num_t_steps, 6]: Body frame of reference (A FoR) acceleration.

Acceleration of the reference frame A in time.

If a case is restarted from a pickle file, the .dyn.h5 file should include the dynamic information for the previous simulation (that will be discarded) and the information for the new simulation.

SHARPy Solvers¶

The available SHARPy solvers are listed below. Attending to SHARPy’s modular structure, they can be run independently so the order in which you desire to run them is important.

The starting point is the PreSharpy loader. It contains the simulation configuration and which solvers are to be run and in the order that should happen.

Aero Solvers¶

DynamicUVLM¶

class sharpy.solvers.dynamicuvlm.DynamicUVLM[source]¶

Dynamic Aerodynamic Time Domain Simulation

Provides an aerodynamic only simulation in time by time stepping the solution. The type of aerodynamic solver is parsed as a setting.

To Do:: Clean timestep information for memory efficiency

Warning

Under development. Issues encountered when using the linear UVLM as the aerodynamic solver with integration order = 1.

The settings that this solver accepts are given by a dictionary, with the following key-value pairs:

Name	Type	Description	Default
`print_info`	`bool`	Write status to screen	`True`
`structural_solver`	`str`	Structural solver to use in the coupled simulation	`None`
`structural_solver_settings`	`dict`	Dictionary of settings for the structural solver	`None`
`aero_solver`	`str`	Aerodynamic solver to use in the coupled simulation	`None`
`aero_solver_settings`	`dict`	Dictionary of settings for the aerodynamic solver	`None`
`n_time_steps`	`int`	Number of time steps for the simulation	`None`
`dt`	`float`	Time step	`None`
`include_unsteady_force_contribution`	`bool`	If on, added mass contribution is added to the forces. This depends on the time derivative of the bound circulation. Check `filter_gamma_dot` in the aero solver	`False`
`postprocessors`	`list(str)`	List of the postprocessors to run at the end of every time step	`[]`
`postprocessors_settings`	`dict`	Dictionary with the applicable settings for every `psotprocessor`. Every `postprocessor` needs its entry, even if empty	`{}`

NoAero¶

class sharpy.solvers.noaero.NoAero[source]¶

Skip UVLM evaluation

Solver to be used with either StaticCoupled or DynamicCoupled when aerodynamics are not of interest.

An example would be running a structural-only simulation where you would like to keep an aerodynamic grid for visualisation purposes.

The settings that this solver accepts are given by a dictionary, with the following key-value pairs:

Name	Type	Description	Default
`dt`	`float`	Time step	`0.1`
`update_grid`	`bool`	Update aerodynamic grid as the structure deforms.	`True`

PrescribedUvlm¶

class sharpy.solvers.prescribeduvlm.PrescribedUvlm[source]¶

This class runs a prescribed rigid body motion simulation of a rigid aerodynamic body.

The settings that this solver accepts are given by a dictionary, with the following key-value pairs:

Name	Type	Description	Default
`print_info`	`bool`	Write status to screen	`True`
`structural_solver`	`str`	Structural solver to use in the coupled simulation	`None`
`structural_solver_settings`	`dict`	Dictionary of settings for the structural solver	`None`
`aero_solver`	`str`	Aerodynamic solver to use in the coupled simulation	`None`
`aero_solver_settings`	`dict`	Dictionary of settings for the aerodynamic solver	`None`
`n_time_steps`	`int`	Number of time steps for the simulation	`None`
`dt`	`float`	Time step	`None`
`postprocessors`	`list(str)`	List of the postprocessors to run at the end of every time step	`[]`
`postprocessors_settings`	`dict`	Dictionary with the applicable settings for every `postprocessor`. Every `postprocessor` needs its entry, even if empty	`{}`
`cfl1`	`bool`	If it is `True`, it assumes that the discretisation complies with CFL=1	`True`
`vortex_radius`	`float`	Distance between points below which induction is not computed	`sharpy.utils.constants.vortex_radius_def`
`vortex_radius_wake_ind`	`float`	Distance between points below which induction is not computed in the wake convection	`sharpy.utils.constants.vortex_radius_def`

StaticUvlm¶

class sharpy.solvers.staticuvlm.StaticUvlm[source]¶

StaticUvlm solver class, inherited from BaseSolver

Aerodynamic solver that runs a UVLM routine to solve the steady or unsteady aerodynamic problem. The aerodynamic problem is posed in the form of an Aerogrid object.

Parameters

data (PreSharpy) – object with problem data
custom_settings (dict) – custom settings that override the settings in the solver .txt file. None by default

settings¶

Name-value pair of settings employed by solver. See Notes for valid combinations

Type: dict

settings_types¶

Acceptable data types for entries in settings

Type: dict

settings_default¶

Default values for the available settings

Type: dict

data¶

object containing the information of the problem

Type: PreSharpy

velocity_generator¶

object containing the flow conditions information

Type: object

The settings that this solver accepts are given by a dictionary, with the following key-value pairs:

Name	Type	Description	Default
`print_info`	`bool`	Print info to screen	`True`
`horseshoe`	`bool`	Horseshoe wake modelling for steady simulations.	`False`
`num_cores`	`int`	Number of cores to use in the VLM lib	`0`
`n_rollup`	`int`	Number of rollup iterations for free wake. Use at least `n_rollup > 1.1*m_star`	`1`
`rollup_dt`	`float`	Controls when the AIC matrix is refreshed during the wake rollup	`0.1`
`rollup_aic_refresh`	`int`		`1`
`rollup_tolerance`	`float`	Convergence criterium for rollup wake	`0.0001`
`iterative_solver`	`bool`	Not in use	`False`
`iterative_tol`	`float`	Not in use	`0.0001`
`iterative_precond`	`bool`	Not in use	`False`
`velocity_field_generator`	`str`	Name of the velocity field generator to be used in the simulation	`SteadyVelocityField`
`velocity_field_input`	`dict`	Dictionary of settings for the velocity field generator	`{}`
`rho`	`float`	Air density	`1.225`
`cfl1`	`bool`	If it is `True`, it assumes that the discretisation complies with CFL=1	`True`
`vortex_radius`	`float`	Distance between points below which induction is not computed	`sharpy.utils.constants.vortex_radius_def`
`vortex_radius_wake_ind`	`float`	Distance between points below which induction is not computed in the wake convection	`sharpy.utils.constants.vortex_radius_def`
`rbm_vel_g`	`list(float)`	Rigid body velocity in G FoR	`[0.0, 0.0, 0.0, 0.0, 0.0, 0.0]`
`centre_rot_g`	`list(float)`	Centre of rotation in G FoR around which `rbm_vel_g` is applied	`[0.0, 0.0, 0.0]`

next_step()[source]¶: Updates de aerogrid based on the info of the step, and increases the self.ts counter

StepLinearUVLM¶

class sharpy.solvers.steplinearuvlm.StepLinearUVLM[source]¶

Time domain aerodynamic solver that uses a linear UVLM formulation to be used with the solvers.DynamicCoupled solver.

To use this solver, the solver_id = StepLinearUVLM must be given as the name for the aero_solver is the case of an aeroelastic solver, where the setting below would be parsed through aero_solver_settings.

Notes

The integr_order variable refers to the finite differencing scheme used to calculate the bound circulation derivative with respect to time $\dot{\mathbf{\Gamma}}$. A first order scheme is used when integr_order == 1

\[\dot{\mathbf{\Gamma}}^{n+1} = \frac{\mathbf{\Gamma}^{n+1}-\mathbf{\Gamma}^n}{\Delta t}\]

If integr_order == 2 a higher order scheme is used (but it isn’t exactly second order accurate [1]).

\[\dot{\mathbf{\Gamma}}^{n+1} = \frac{3\mathbf{\Gamma}^{n+1}-4\mathbf{\Gamma}^n + \mathbf{\Gamma}^{n-1}} {2\Delta t}\]

If track_body is True, the UVLM is projected onto a frame U that is:

Coincident with G at the linearisation timestep.

Thence, rotates by the same quantity as the FoR A.

It is similar to a stability axes and is recommended any time rigid body dynamics are included.

See also

sharpy.sharpy.linear.assembler.linearuvlm.LinearUVLM

References

[1] Maraniello, S., & Palacios, R.. State-Space Realizations and Internal Balancing in Potential-Flow Aerodynamics with Arbitrary Kinematics. AIAA Journal, 57(6), 1–14. 2019. https://doi.org/10.2514/1.J058153

The settings that this solver accepts are given by a dictionary, with the following key-value pairs:

Name	Type	Description	Default
`dt`	`float`	Time step	`0.1`
`integr_order`	`int`	Integration order of the circulation derivative. Either `1` or `2`.	`2`
`ScalingDict`	`dict`	Dictionary of scaling factors to achieve normalised UVLM realisation.	`{}`
`remove_predictor`	`bool`	Remove the predictor term from the UVLM equations	`True`
`use_sparse`	`bool`	Assemble UVLM plant matrix in sparse format	`True`
`density`	`float`	Air density	`1.225`
`track_body`	`bool`	UVLM inputs and outputs projected to coincide with lattice at linearisation	`True`
`track_body_number`	`int`	Frame of reference number to follow. If `-1` track `A` frame.	`-1`
`velocity_field_generator`	`str`	Name of the velocity field generator to be used in the simulation	`SteadyVelocityField`
`velocity_field_input`	`dict`	Dictionary of settings for the velocity field generator	`{}`
`vortex_radius`	`float`	Distance between points below which induction is not computed	`sharpy.utils.constants.vortex_radius_def`
`vortex_radius_wake_ind`	`float`	Distance between points below which induction is not computed in the wake convection	`sharpy.utils.constants.vortex_radius_def`
`cfl1`	`bool`	If it is `True`, it assumes that the discretisation complies with CFL=1	`True`

The settings that ScalingDict accepts are the following:

Name	Type	Description	Default
`length`	`float`	Reference length to be used for UVLM scaling	`1.0`
`speed`	`float`	Reference speed to be used for UVLM scaling	`1.0`
`density`	`float`	Reference density to be used for UVLM scaling	`1.0`

initialise(data, custom_settings=None)[source]¶

Initialises the Linear UVLM aerodynamic solver and the chosen velocity generator.

Settings are parsed into the standard SHARPy settings format for solvers. It then checks whether there is any previous information about the linearised system (in order for a solution to be restarted without overwriting the linearisation).

If a linearised system does not exist, a linear UVLM system is created linearising about the current time step.

The reference values for the input and output are transformed into column vectors $\mathbf{u}$ and $\mathbf{y}$, respectively.

The information pertaining to the linear system is stored in a dictionary self.data.aero.linear within the main data variable.

Parameters

data (PreSharpy) – class containing the problem information
custom_settings (dict) – custom settings dictionary

pack_input_vector()[source]¶

Transform a SHARPy AeroTimestep instance into a column vector containing the input to the linear UVLM system.

\[[\zeta,\, \dot{\zeta}, u_{ext}] \longrightarrow \mathbf{u}\]

If the track_body option is on, the function projects all the input into a frame that:

is equal to the FoR G at time 0 (linearisation point)

rotates as the body frame specified in the track_body_number

Returns: Input vector
Return type: np.ndarray

static pack_state_vector(aero_tstep, aero_tstep_m1, dt, integr_order)[source]¶

Transform SHARPy Aerotimestep format into column vector containing the state information.

The state vector is of a different form depending on the order of integration chosen. If a second order scheme is chosen, the state includes the bound circulation at the previous timestep, hence the timestep information for the previous timestep shall be parsed.

The transformation is of the form:

If integr_order==1:

\[\mathbf{x}_n = [\mathbf{\Gamma}^T_n,\, \mathbf{\Gamma_w}_n^T,\, \Delta t \,\mathbf{\dot{\Gamma}}_n^T]^T\]
Else, if integr_order==2:

\[\mathbf{x}_n = [\mathbf{\Gamma}_n^T,\, \mathbf{\Gamma_w}_n^T,\, \Delta t \,\mathbf{\dot{\Gamma}}_n^T,\, \mathbf{\Gamma}_{n-1}^T]^T\]

For the second order integration scheme, if the previous timestep information is not parsed, a first order stencil is employed to estimate the bound circulation at the previous timestep:

\[\mathbf{\Gamma}^{n-1} = \mathbf{\Gamma}^n - \Delta t \mathbf{\dot{\Gamma}}^n\]

Parameters

aero_tstep (AeroTimeStepInfo) – Aerodynamic timestep information at the current timestep n.
aero_tstep_m1 (AeroTimeStepInfo) –

Returns

State vector

Return type

np.ndarray

run(aero_tstep, structure_tstep, convect_wake=False, dt=None, t=None, unsteady_contribution=False)[source]¶

Solve the linear aerodynamic UVLM model at the current time step n. The step increment is solved as:

\[\begin{split}\mathbf{x}^n &= \mathbf{A\,x}^{n-1} + \mathbf{B\,u}^n \\ \mathbf{y}^n &= \mathbf{C\,x}^n + \mathbf{D\,u}^n\end{split}\]

A change of state is possible in order to solve the system without the predictor term. In which case the system is solved by:

\[\begin{split}\mathbf{h}^n &= \mathbf{A\,h}^{n-1} + \mathbf{B\,u}^{n-1} \\ \mathbf{y}^n &= \mathbf{C\,h}^n + \mathbf{D\,u}^n\end{split}\]

Variations are taken with respect to initial reference state. The state and input vectors for the linear UVLM system are of the form:

If integr_order==1:
\[\mathbf{x}_n = [\delta\mathbf{\Gamma}^T_n,\, \delta\mathbf{\Gamma_w}_n^T,\, \Delta t \,\delta\mathbf{\dot{\Gamma}}_n^T]^T\]

Else, if integr_order==2:
\[\mathbf{x}_n = [\delta\mathbf{\Gamma}_n^T,\, \delta\mathbf{\Gamma_w}_n^T,\, \Delta t \,\delta\mathbf{\dot{\Gamma}}_n^T,\, \delta\mathbf{\Gamma}_{n-1}^T]^T\]

And the input vector:
\[\mathbf{u}_n = [\delta\mathbf{\zeta}_n^T,\, \delta\dot{\mathbf{\zeta}}_n^T,\,\delta\mathbf{u_{ext}}^T_n]^T\]

where the subscript n refers to the time step.

The linear UVLM system is then solved as detailed in sharpy.linear.src.linuvlm.Dynamic.solve_step(). The output is a column vector containing the aerodynamic forces at the panel vertices.

To Do: option for impulsive start?

Parameters

aero_tstep (AeroTimeStepInfo) – object containing the aerodynamic data at the current time step
structure_tstep (StructTimeStepInfo) – object containing the structural data at the current time step
convect_wake (bool) – for backward compatibility only. The linear UVLM assumes a frozen wake geometry
dt (float) – time increment
t (float) – current time
unsteady_contribution (bool) – (backward compatibily). Unsteady aerodynamic effects are always included

Returns

updated self.data class with the new forces and circulation terms of the system

Return type

PreSharpy

unpack_ss_vectors(y_n, x_n, u_n, aero_tstep)[source]¶

Transform column vectors used in the state space formulation into SHARPy format

The column vectors are transformed into lists with one entry per aerodynamic surface. Each entry contains a matrix with the quantities at each grid vertex.

\[\mathbf{y}_n \longrightarrow \mathbf{f}_{aero}\]

\[\mathbf{x}_n \longrightarrow \mathbf{\Gamma}_n,\, \mathbf{\Gamma_w}_n,\, \mathbf{\dot{\Gamma}}_n\]

If the track_body option is on, the output forces are projected from the linearization frame, to the G frame. Note that the linearisation frame is:

equal to the FoR G at time 0 (linearisation point)

rotates as the body frame specified in the track_body_number

Parameters

y_n (np.ndarray) – Column output vector of linear UVLM system
x_n (np.ndarray) – Column state vector of linear UVLM system
u_n (np.ndarray) – Column input vector of linear UVLM system
aero_tstep (AeroTimeStepInfo) – aerodynamic timestep information class instance

Returns

Tuple containing:

forces (list):
Aerodynamic forces in a list with n_surf entries. Each entry is a (6, M+1, N+1) matrix, where the first 3 indices correspond to the components in x, y and z. The latter 3 are zero.

gamma (list):
Bound circulation list with n_surf entries. Circulation is stored in an (M+1, N+1) matrix, corresponding to the panel vertices.

gamma_dot (list):
Bound circulation derivative list with n_surf entries. Circulation derivative is stored in an (M+1, N+1) matrix, corresponding to the panel vertices.

gamma_star (list):
Wake (free) circulation list with n_surf entries. Wake circulation is stored in an (M_star+1, N+1) matrix, corresponding to the panel vertices of the wake.

Return type

tuple

StepUvlm¶

class sharpy.solvers.stepuvlm.StepUvlm[source]¶

StepUVLM is the main solver to use for unsteady aerodynamics.

The desired flow field is injected into the simulation by means of a generator. For a list of available velocity field generators see the documentation page on generators which can be found under SHARPy Source Code.

Typical generators could be:

SteadyVelocityField
GustVelocityField
TurbVelocityField

amongst others.

The settings that this solver accepts are given by a dictionary, with the following key-value pairs:

Name	Type	Description	Default	Options
`print_info`	`bool`	Print info to screen	`True`
`num_cores`	`int`	Number of cores to use in the VLM lib	`0`
`n_time_steps`	`int`	Number of time steps to be run	`100`
`convection_scheme`	`int`	`0`: fixed wake, `2`: convected with background flow;``3``: full force-free wake	`3`	`0`, `2`, `3`
`dt`	`float`	Time step	`0.1`
`iterative_solver`	`bool`	Not in use	`False`
`iterative_tol`	`float`	Not in use	`0.0001`
`iterative_precond`	`bool`	Not in use	`False`
`velocity_field_generator`	`str`	Name of the velocity field generator to be used in the simulation	`SteadyVelocityField`
`velocity_field_input`	`dict`	Dictionary of settings for the velocity field generator	`{}`
`gamma_dot_filtering`	`int`	Filtering parameter for the Welch filter for the Gamma_dot estimation. Used when `unsteady_force_contribution` is `on`.	`0`
`rho`	`float`	Air density	`1.225`
`cfl1`	`bool`	If it is `True`, it assumes that the discretisation complies with CFL=1	`True`
`vortex_radius`	`float`	Distance between points below which induction is not computed	`sharpy.utils.constants.vortex_radius_def`
`vortex_radius_wake_ind`	`float`	Distance between points below which induction is not computed in the wake convection	`sharpy.utils.constants.vortex_radius_def`
`interp_coords`	`int`	Coordinates to use for wake description: cartesian(0) or cylindrical_z(1)	`0`	`0`, `1`
`filter_method`	`int`	Method to filter the points: no filter (0) moving average(2)	`0`	`0`, `2`
`interp_method`	`int`	Method of interpolation: linear(0), parabolic(1), splines(2), slerp around z(3), slerp around yaw_slerp(4)	`0`	`0`, `1`, `2`, `3`, `4`
`yaw_slerp`	`float`	Yaw angle in radians to be used when interp_metod == 4	`0`
`centre_rot`	`list(float)`	Coordinates of the centre of rotation to perform slerp interpolation or cylindrical coordinates	`[0.0, 0.0, 0.0]`
`quasi_steady`	`bool`	Use quasi-steady approximation in UVLM	`False`

initialise(data, custom_settings=None)[source]¶: To be called just once per simulation.

run(aero_tstep=None, structure_tstep=None, convect_wake=True, dt=None, t=None, unsteady_contribution=False)[source]¶: Runs a step of the aerodynamics as implemented in UVLM.

Coupled Solvers¶

DynamicCoupled¶

class sharpy.solvers.dynamiccoupled.DynamicCoupled[source]¶

The DynamicCoupled solver couples the aerodynamic and structural solvers of choice to march forward in time the aeroelastic system’s solution.

Using the DynamicCoupled solver requires that an instance of the StaticCoupled solver is called in the SHARPy solution flow when defining the problem case.

Input data (from external controllers) can be received and data sent using the SHARPy network interface, specified through the setting network_settings of this solver. For more detail on how to send and receive data see the NetworkLoader documentation.

Changes to the structural properties or external forces that depend on the instantaneous situation of the system can be applied through runtime_generators. These runtime generators are parsed through dictionaries, with the key being the name of the generator and the value the settings for such generator. The currently available runtime_generators are ExternalForces and ModifyStructure.

The settings that this solver accepts are given by a dictionary, with the following key-value pairs:

Name	Type	Description	Default	Options
`print_info`	`bool`	Write status to screen	`True`
`structural_solver`	`str`	Structural solver to use in the coupled simulation	`None`
`structural_solver_settings`	`dict`	Dictionary of settings for the structural solver	`None`
`aero_solver`	`str`	Aerodynamic solver to use in the coupled simulation	`None`
`aero_solver_settings`	`dict`	Dictionary of settings for the aerodynamic solver	`None`
`n_time_steps`	`int`	Number of time steps for the simulation	`None`
`dt`	`float`	Time step	`None`
`fsi_substeps`	`int`	Max iterations in the FSI loop	`70`
`fsi_tolerance`	`float`	Convergence threshold for the FSI loop	`1e-05`
`structural_substeps`	`int`	Number of extra structural time steps per aero time step. `0` is a fully coupled simulation.	`0`
`relaxation_factor`	`float`	Relaxation parameter in the FSI iteration. `0` is no relaxation and -> `1` is very relaxed	`0.2`
`final_relaxation_factor`	`float`	Relaxation factor reached in `relaxation_steps` with `dynamic_relaxation` on	`0.0`
`minimum_steps`	`int`	Number of minimum FSI iterations before convergence	`3`
`relaxation_steps`	`int`	Length of the relaxation factor ramp between `relaxation_factor` and `final_relaxation_factor` with `dynamic_relaxation` on	`100`
`dynamic_relaxation`	`bool`	Controls if relaxation factor is modified during the FSI iteration process	`False`
`postprocessors`	`list(str)`	List of the postprocessors to run at the end of every time step	`[]`
`postprocessors_settings`	`dict`	Dictionary with the applicable settings for every `postprocessor`. Every `postprocessor` needs its entry, even if empty	`{}`
`controller_id`	`dict`	Dictionary of id of every controller (key) and its type (value)	`{}`
`controller_settings`	`dict`	Dictionary with settings (value) of every controller id (key)	`{}`
`cleanup_previous_solution`	`bool`	Controls if previous `timestep_info` arrays are reset before running the solver	`False`
`include_unsteady_force_contribution`	`bool`	If on, added mass contribution is added to the forces. This depends on the time derivative of the bound circulation. Check `filter_gamma_dot` in the aero solver	`False`
`steps_without_unsteady_force`	`int`	Number of initial timesteps that don’t include unsteady forces contributions. This avoids oscillations due to no perfectly trimmed initial conditions	`0`
`pseudosteps_ramp_unsteady_force`	`int`	Length of the ramp with which unsteady force contribution is introduced every time step during the FSI iteration process	`0`
`correct_forces_method`	`str`	Function used to correct aerodynamic forces. See `sharpy.generators.polaraeroforces`		`EfficiencyCorrection`, `PolarCorrection`
`correct_forces_settings`	`dict`	Settings for corrected forces evaluation	`{}`
`network_settings`	`dict`	Network settings. See `NetworkLoader` for supported entries	`{}`
`runtime_generators`	`dict`	The dictionary keys are the runtime generators to be used. The dictionary values are dictionaries with the settings needed by each generator.	`{}`

convergence(k, tstep, previous_tstep, struct_solver, aero_solver, with_runtime_generators)[source]¶

Check convergence in the FSI loop.

Convergence is determined as:

\[\epsilon_q^k = \frac{|| q^k - q^{k - 1} ||}{q^0}\]

\[\epsilon_\dot{q}^k = \frac{|| \dot{q}^k - \dot{q}^{k - 1} ||}{\dot{q}^0}\]

FSI converged if $\epsilon_q^k < \mathrm{FSI\ tolerance}$ and $\epsilon_\dot{q}^k < \mathrm{FSI\ tolerance}$

get_g()[source]¶: Getter for g, the gravity value

get_rho()[source]¶: Getter for rho, the density value

initialise(data, custom_settings=None)[source]¶: Controls the initialisation process of the solver, including processing the settings and initialising the aero and structural solvers, postprocessors and controllers.

static interpolate_timesteps(step0, step1, out_step, coeff)[source]¶

Performs a linear interpolation between step0 and step1 based on coeff in [0, 1]. 0 means info in out_step == step0 and 1 out_step == step1.

Quantities interpolated: * steady_applied_forces * unsteady_applied_forces * velocity input in Lagrange constraints

process_controller_output(controlled_state)[source]¶

This function modified the solver properties and parameters as requested from the controller.

This keeps the main loop much cleaner, while allowing for flexibility

Please, if you add options in here, always code the possibility of that specific option not being there without the code complaining to the user.

If it possible, use the same Key for the new setting as for the setting in the solver. For example, if you want to modify the structural_substeps variable in settings, use that Key in the info dictionary.

As a convention: a value of None returns the value to the initial one specified in settings, while the key not being in the dict is ignored, so if any change was made before, it will stay there.

run()[source]¶: Run the time stepping procedure with controllers and postprocessors included.

set_g(new_g)[source]¶: Setter for g, the gravity value

set_rho(new_rho)[source]¶: Setter for rho, the density value

LinDynamicSim¶

class sharpy.solvers.lindynamicsim.LinDynamicSim[source]¶

Time-domain solution of Linear Time Invariant Systems

Uses the derived linear time invariant systems and solves it in time domain.

Requires a case_name.lininput.h5 file in the case root folder that contains:

x0 (optional): Initial state vector

input_vec: Input vector (n_tsteps, n_inputs).

Note

This solver is seldom used in SHARPy (its focus is on nonlinear time domain aeroelasticity) hence you may find this solver lacking in features. If you use it, you may need to make modifications. We would greatly appreciate that you contribute these modifications by means of a pull request!

The settings that this solver accepts are given by a dictionary, with the following key-value pairs:

Name	Type	Description	Default
`write_dat`	`list(str)`	List of vectors to write: `x`, `y`, `u` and/or `t`	`[]`
`reference_velocity`	`float`	Velocity to scale the structural equations when using a non-dimensional system	`1.0`
`n_tsteps`	`int`	Number of time steps to run	`10`
`physical_time`	`float`	Time to run	`2.0`
`dt`	`float`	Time increment for the solution of systems without a specified dt	`0.001`
`postprocessors`	`list(str)`		`[]`
`postprocessors_settings`	`dict`		`{}`

StaticCoupled¶

class sharpy.solvers.staticcoupled.StaticCoupled[source]¶

This class is the main FSI driver for static simulations. It requires a structural_solver and a aero_solver to be defined.

The settings that this solver accepts are given by a dictionary, with the following key-value pairs:

Name	Type	Description	Default	Options
`print_info`	`bool`	Write status to screen	`True`
`structural_solver`	`str`	Structural solver to use in the coupled simulation	`None`
`structural_solver_settings`	`dict`	Dictionary of settings for the structural solver	`None`
`aero_solver`	`str`	Aerodynamic solver to use in the coupled simulation	`None`
`aero_solver_settings`	`dict`	Dictionary of settings for the aerodynamic solver	`None`
`max_iter`	`int`	Max iterations in the FSI loop	`100`
`n_load_steps`	`int`	Length of ramp for forces and gravity during FSI iteration	`0`
`tolerance`	`float`	Convergence threshold for the FSI loop	`1e-05`
`relaxation_factor`	`float`	Relaxation parameter in the FSI iteration. 0 is no relaxation and -> 1 is very relaxed	`0.0`
`correct_forces_method`	`str`	Function used to correct aerodynamic forces. See `sharpy.generators.polaraeroforces`		`EfficiencyCorrection`, `PolarCorrection`
`correct_forces_settings`	`dict`	Settings for corrected forces evaluation	`{}`
`runtime_generators`	`dict`	The dictionary keys are the runtime generators to be used. The dictionary values are dictionaries with the settings needed by each generator.	`{}`

StaticCoupledRBM¶

class sharpy.solvers.staticcoupledrbm.StaticCoupledRBM[source]¶

Steady coupled solver including rigid body motions

The settings that this solver accepts are given by a dictionary, with the following key-value pairs:

Name	Type	Description	Default	Options
`print_info`	`bool`	Output run-time information	`True`
`structural_solver`	`str`	Name of the structural solver used in the computation	`None`
`structural_solver_settings`	`dict`	Dictionary os settings needed by the structural solver	`None`
`aero_solver`	`str`	Name of the aerodynamic solver used in the computation	`None`
`aero_solver_settings`	`dict`	Dictionary os settings needed by the aerodynamic solver	`None`
`max_iter`	`int`	Maximum numeber of FSI iterations	`100`
`n_load_steps`	`int`	Number of steps to ramp up the application of loads	`1`
`tolerance`	`float`	FSI tolerance	`1e-05`
`relaxation_factor`	`float`	Relaxation factor	`0.0`
`correct_forces_method`	`str`	Function used to correct aerodynamic forces. See `sharpy.generators.polaraeroforces`		`EfficiencyCorrection`, `PolarCorrection`
`correct_forces_settings`	`dict`	Settings for corrected forces evaluation	`{}`

Flight dynamics Solvers¶

StaticTrim¶

class sharpy.solvers.statictrim.StaticTrim[source]¶

The StaticTrim solver determines the longitudinal state of trim (equilibrium) for an aeroelastic system in static conditions. It wraps around the desired solver to yield the state of trim of the system, in most cases the StaticCoupled solver.

It calculates the required angle of attack, elevator deflection and thrust required to achieve longitudinal equilibrium. The output angles are shown in degrees.

The results from the trimming iteration can be saved to a text file by using the save_info option.

The settings that this solver accepts are given by a dictionary, with the following key-value pairs:

Name	Type	Description	Default
`print_info`	`bool`	Print info to screen	`True`
`solver`	`str`	Solver to run in trim routine
`solver_settings`	`dict`	Solver settings dictionary	`{}`
`max_iter`	`int`	Maximum number of iterations of trim routine	`100`
`fz_tolerance`	`float`	Tolerance in vertical force	`0.01`
`fx_tolerance`	`float`	Tolerance in horizontal force	`0.01`
`m_tolerance`	`float`	Tolerance in pitching moment	`0.01`
`tail_cs_index`	`int`	Index of control surfaces that move to achieve trim	`0`
`thrust_nodes`	`list(int)`	Nodes at which thrust is applied	`[0]`
`initial_alpha`	`float`	Initial angle of attack	`0.0`
`initial_deflection`	`float`	Initial control surface deflection	`0.0`
`initial_thrust`	`float`	Initial thrust setting	`0.0`
`initial_angle_eps`	`float`	Initial change of control surface deflection	`0.05`
`initial_thrust_eps`	`float`	Initial thrust setting change	`2.0`
`relaxation_factor`	`float`	Relaxation factor	`0.2`
`save_info`	`bool`	Save trim results to text file	`False`

trim_algorithm()[source]¶

Trim algorithm method

The trim condition is found iteratively.

Returns: array of trim values for angle of attack, control surface deflection and thrust.
Return type: np.array

Trim¶

class sharpy.solvers.trim.Trim[source]¶

Trim routine with support for lateral dynamics. It usually struggles much more than the StaticTrim (only longitudinal) solver.

We advise to start with StaticTrim even if you configuration is not totally symmetric.

The settings that this solver accepts are given by a dictionary, with the following key-value pairs:

Name	Type	Description	Default
`print_info`	`bool`	Print info to screen	`True`
`solver`	`str`	Solver to run in trim routine
`solver_settings`	`dict`	Solver settings dictionary	`{}`
`max_iter`	`int`	Maximum number of iterations of trim routine	`100`
`tolerance`	`float`	Threshold for convergence of trim	`0.0001`
`initial_alpha`	`float`	Initial angle of attack	`0.0`
`initial_beta`	`float`	Initial sideslip angle	`0.0`
`initial_roll`	`float`	Initial roll angle	`0`
`cs_indices`	`list(int)`	Indices of control surfaces to be trimmed	`[]`
`initial_cs_deflection`	`list(float)`	Initial deflection of the control surfaces in order.	`[]`
`thrust_nodes`	`list(int)`	Nodes at which thrust is applied	`[0]`
`initial_thrust`	`list(float)`	Initial thrust setting	`[1.0]`
`thrust_direction`	`list(float)`	Thrust direction setting	`[0.0, 1.0, 0.0]`
`special_case`	`dict`	Extra settings for specific cases such as differential thrust control	`{}`
`refine_solution`	`bool`	If `True` and the optimiser routine allows for it, the optimiser will try to improve the solution with hybrid methods	`False`

Linear Solvers¶

LinearAssembler¶

class sharpy.solvers.linearassembler.LinearAssembler[source]¶

Warning

Under development - please advise of new features and bugs!

Creates a workspace containing the different linear elements of the state-space.

The user specifies which elements to build sequentially via the linear_system setting.

The most common uses will be:

Aerodynamic: sharpy.linear.assembler.LinearUVLM solver

Structural: sharpy.linear.assembler.LinearBeam solver

Aeroelastic: sharpy.linear.assembler.LinearAeroelastic solver

The solver enables to load a user specific assembly of a state-space by means of the LinearCustom block.

See sharpy.sharpy.linear.assembler.LinearAssembler for a detailed description of each of the state-space assemblies.

Upon assembly of the linear system, the data structure data.linear will be created. The Linear contains the state-space as an attribute. This state space will be the one employed by postprocessors.

Important: running the linear routines requires information on the tangent mass, stiffness and gyroscopic structural matrices therefore the solver solvers.modal.Modal must have been run prior to linearisation. In addition, if the problem includes rigid body velocities, at least one timestep of solvers.DynamicCoupled must have run such that the rigid body velocity is included.

Example:

The typical flow setting used prior to using this solver for an aeroelastic simulation with rigid body dynamics will be similar to:

>>> flow = ['BeamLoader',
>>>        'AerogridLoader',
>>>        'StaticTrim',
>>>        'DynamicCoupled',  # a single time step will suffice
>>>        'Modal',
>>>        'LinearAssembler']

The settings that this solver accepts are given by a dictionary, with the following key-value pairs:

Name	Type	Description	Default	Options
`linear_system`	`str`	Name of chosen state space assembly type	`None`
`linear_system_settings`	`dict`	Settings for the desired state space assembler	`{}`
`linearisation_tstep`	`int`	Chosen linearisation time step number from available time steps	`-1`
`modal_tstep`	`int`	Timestep in which modal information is stored. Useful if the `Modal` solver is run at the start of the SHARPy flow.	`-1`
`inout_coordinates`	`str`	Input/output coordinates of the system. Nodal or modal space.		``, ``nodes, `modes`
`retain_inputs`	`list(int)`	List of input channels to retain in the chosen `inout_coordinates`.	`[]`
`retain_outputs`	`list(int)`	List of output channels to retain in the chosen `inout_coordinates`.	`[]`

Modal¶

class sharpy.solvers.modal.Modal[source]¶

Modal solver class, inherited from BaseSolver

Extracts the M, K and C matrices from the Fortran library for the beam. Depending on the choice of modal projection, these may or may not be transformed to a state-space form to compute the eigenvalues and mode shapes of the structure.

The settings that this solver accepts are given by a dictionary, with the following key-value pairs:

Name	Type	Description	Default
`print_info`	`bool`	Write status to screen	`True`
`rigid_body_modes`	`bool`	Write modes with rigid body mode shapes	`False`
`use_undamped_modes`	`bool`	Project the modes onto undamped mode shapes	`True`
`NumLambda`	`int`	Number of modes to retain	`20`
`keep_linear_matrices`	`bool`	Save M, C and K matrices to output dictionary	`True`
`write_modes_vtk`	`bool`	Write Paraview files with mode shapes	`True`
`print_matrices`	`bool`	Write M, C and K matrices to file	`False`
`write_dat`	`bool`	Write mode shapes, frequencies and damping to file	`True`
`continuous_eigenvalues`	`bool`	Use continuous time eigenvalues	`False`
`dt`	`float`	Time step to compute discrete time eigenvalues	`0`
`delta_curved`	`float`	Threshold for linear expressions in rotation formulas	`0.01`
`plot_eigenvalues`	`bool`	Plot to screen root locus diagram	`False`
`max_rotation_deg`	`float`	Scale mode shape to have specified maximum rotation	`15.0`
`max_displacement`	`float`	Scale mode shape to have specified maximum displacement	`0.15`
`use_custom_timestep`	`int`	If > -1, it will use that time step geometry for calculating the modes	`-1`
`rigid_modes_cg`	`bool`	Modify the ridid body modes such that they are defined wrt to the CG	`False`

free_free_modes(phi, M)[source]¶

Warning

This function is deprecated. See free_modes_principal_axes() for a transformation to the CG and with respect to the principal axes of inertia.

Returns the rigid body modes defined with respect to the centre of gravity

The transformation from the modes defined at the FoR A origin, $\boldsymbol{\Phi}$, to the modes defined using the centre of gravity as a reference is

\[\boldsymbol{\Phi}_{rr,CG}|_{TRA} = \boldsymbol{\Phi}_{RR}|_{TRA} + \tilde{\mathbf{r}}_{CG} \boldsymbol{\Phi}_{RR}|_{ROT}\]

\[\boldsymbol{\Phi}_{rr,CG}|_{ROT} = \boldsymbol{\Phi}_{RR}|_{ROT}\]

Returns: Transformed eigenvectors
Return type: (np.array)

run()[source]¶

Extracts the eigenvalues and eigenvectors of the clamped structure.

If use_undamped_modes == True then the free vibration modes of the clamped structure are found solving:

\[\mathbf{M\,\ddot{\eta}} + \mathbf{K\,\eta} = 0\]

that flows down to solving the non-trivial solutions to:

\[(-\omega_n^2\,\mathbf{M} + \mathbf{K})\mathbf{\Phi} = 0\]

On the other hand, if the damped modes are chosen because the system has damping, the free vibration modes are found solving the equation of motion of the form:

\[\mathbf{M\,\ddot{\eta}} + \mathbf{C\,\dot{\eta}} + \mathbf{K\,\eta} = 0\]

which can be written in state space form, with the state vector $\mathbf{x} = [\eta^T,\,\dot{\eta}^T]^T$ as

\[\begin{split}\mathbf{\dot{x}} = \begin{bmatrix} 0 & \mathbf{I} \\ -\mathbf{M^{-1}K} & -\mathbf{M^{-1}C} \end{bmatrix} \mathbf{x}\end{split}\]

and therefore the mode shapes and frequencies correspond to the solution of the eigenvalue problem

\[\mathbf{A\,\Phi} = \mathbf{\Lambda\,\Phi}.\]

From the eigenvalues, the following system characteristics are provided:

Natural Frequency: $\omega_n = |\lambda|$

Damped natural frequency: $\omega_d = \text{Im}(\lambda) = \omega_n \sqrt{1-\zeta^2}$

Damping ratio: $\zeta = -\frac{\text{Re}(\lambda)}{\omega_n}$

In addition to the above, the modal output dictionary includes the following:

M: Tangent mass matrix

C: Tangent damping matrix

K: Tangent stiffness matrix

Ccut: Modal damping matrix $\mathbf{C}_m = \mathbf{\Phi}^T\mathbf{C}\mathbf{\Phi}$

Kin_damp: Forces gain matrix (when damped): $K_{in} = \mathbf{\Phi}_L^T \mathbf{M}^{-1}$

eigenvectors: Right eigenvectors

eigenvectors_left: Left eigenvectors given when the system is damped

Returns: updated data object with modal analysis as part of the last structural time step.
Return type: PreSharpy

Loader Solvers¶

PreSharpy¶

AerogridLoader¶

class sharpy.solvers.aerogridloader.AerogridLoader[source]¶

AerogridLoader class, inherited from BaseSolver

Generates aerodynamic grid based on the input data

The initial wake shape is now defined in SHARPy (instead of UVLM) through a wake shape generator wake_shape_generator and the required inputs wake_shape_generator_input. The supported wake generators are sharpy.generators.straightwake.StraighWake and sharpy.generators.helicoidalwake.HelicoidalWake.

The control_surface_deflection setting allows the user to use a time specific control surface deflection, should the problem include them. This setting takes a list of strings, each for the required control surface generator.

The control_surface_deflection_generator_settings setting is a list of dictionaries, one for each control surface. The dictionaries specify the settings for the generator DynamicControlSurface. If the relevant control surface is simply static, an empty string should be parsed. See the documentation for DynamicControlSurface generators for accepted key-value pairs as settings.

Parameters: data (PreSharpy) – ProblemData class structure

settings¶

Name-value pair of the settings employed by the aerodynamic solver

Type: dict

settings_types¶

Acceptable types for the values in settings

Type: dict

settings_default¶

Name-value pair of default values for the aerodynamic settings

Type: dict

data¶

class structure

Type: ProblemData

aero_file_name¶

name of the .aero.h5 HDF5 file

Type: str

aero¶: empty attribute

aero_data_dict¶

key-value pairs of aerodynamic data

Type: dict

wake_shape_generator¶

Wake shape generator

Type: class

The settings that this solver accepts are given by a dictionary, with the following key-value pairs:

Name	Type	Description	Default	Options
`unsteady`	`bool`	Unsteady effects	`False`
`aligned_grid`	`bool`	Align grid	`True`
`freestream_dir`	`list(float)`	Free stream flow direction	`[1.0, 0.0, 0.0]`
`mstar`	`int`	Number of chordwise wake panels	`10`
`control_surface_deflection`	`list(str)`	List of control surface generators for each control surface	`[]`
`control_surface_deflection_generator_settings`	`dict`	List of dictionaries with the settings for each generator	`{}`
`wake_shape_generator`	`str`	ID of the generator to define the initial wake shape	`StraightWake`	`StraightWake`, `HelicoidalWake`
`wake_shape_generator_input`	`dict`	Dictionary of inputs needed by the wake shape generator	`{}`

BeamLoader¶

class sharpy.solvers.beamloader.BeamLoader[source]¶

BeamLoader class solver inherited from BaseSolver

Loads the structural beam solver with the specified user settings.

Parameters: data (ProblemData) – class containing the problem information

settings¶

contains the specific settings for the solver

Type: dict

settings_types¶

Key value pairs of the accepted types for the settings values

Type: dict

settings_default¶

Dictionary containing the default solver settings, should none be provided.

Type: dict

data¶

class containing the data for the problem

Type: ProblemData

fem_file_name¶

name of the .fem.h5 HDF5 file

Type: str

dyn_file_name¶

name of the .dyn.h5 HDF5 file

Type: str

fem_data_dict¶

key-value pairs of FEM data

Type: dict

dyn_data_dict¶

key-value pairs of data for dynamic problems

Type: dict

structure¶

Empty attribute

Type: None

Notes

For further reference on Quaternions see: https://en.wikipedia.org/wiki/Quaternion

See also

class sharpy.utils.solver_interface.BaseSolver¶

class sharpy.structure.models.beam.Beam¶

The settings that this solver accepts are given by a dictionary, with the following key-value pairs:

Name	Type	Description	Default
`unsteady`	`bool`	If `True` it will be a dynamic problem and the solver will look for the `.dyn.h5` file that contains the time varying input to the problem.	`True`
`orientation`	`list(float)`	Initial attitude of the structure given as the quaternion that parametrises the rotation from G to A frames of reference.	`[1.0, 0, 0, 0]`
`for_pos`	`list(float)`	Initial position of the A FoR.	`[0.0, 0, 0]`

Structural Solvers¶

NonLinearDynamic¶

class sharpy.solvers.nonlineardynamic.NonLinearDynamic[source]¶

Structural solver used for the dynamic simulation of free-flying structures.

This solver provides an interface to the structural library (xbeam) and updates the structural parameters for every time step of the simulation.

This solver is called as part of a standalone structural simulation.

The settings that this solver accepts are given by a dictionary, with the following key-value pairs:

Name	Type	Description	Default
`print_info`	`bool`	Print output to screen	`True`
`max_iterations`	`int`	Sets maximum number of iterations	`100`
`num_load_steps`	`int`		`1`
`delta_curved`	`float`		`0.01`
`min_delta`	`float`	Structural solver tolerance	`1e-05`
`newmark_damp`	`float`	Sets the Newmark damping coefficient	`0.0001`
`gravity_on`	`bool`	Flag to include gravitational forces	`False`
`gravity`	`float`	Gravitational acceleration	`9.81`
`gravity_dir`	`list(float)`	Direction in G where gravity applies	`[0, 0, 1]`
`relaxation_factor`	`float`		`0.3`
`dt`	`float`	Time step increment	`0.01`
`num_steps`	`int`		`500`
`prescribed_motion`	`bool`		`None`

NonLinearDynamicCoupledStep¶

class sharpy.solvers.nonlineardynamiccoupledstep.NonLinearDynamicCoupledStep[source]¶

Structural solver used for the dynamic simulation of free-flying structures.

This solver provides an interface to the structural library (xbeam) and updates the structural parameters for every k-th step in the FSI iteration.

This solver can be called as part of a standalone structural simulation or as the structural solver of a coupled aeroelastic simulation.

The settings that this solver accepts are given by a dictionary, with the following key-value pairs:

Name	Type	Description	Default
`print_info`	`bool`	Print output to screen	`True`
`max_iterations`	`int`	Sets maximum number of iterations	`100`
`num_load_steps`	`int`		`1`
`delta_curved`	`float`		`0.01`
`min_delta`	`float`	Structural solver tolerance	`1e-05`
`newmark_damp`	`float`	Sets the Newmark damping coefficient	`0.0001`
`gravity_on`	`bool`	Flag to include gravitational forces	`False`
`gravity`	`float`	Gravitational acceleration	`9.81`
`gravity_dir`	`list(float)`	Direction in G where gravity applies	`[0.0, 0.0, 1.0]`
`relaxation_factor`	`float`		`0.3`
`dt`	`float`	Time step increment	`0.01`
`num_steps`	`int`		`500`
`balancing`	`bool`		`False`
`initial_velocity_direction`	`list(float)`	Initial velocity of the reference node given in the inertial FOR	`[-1.0, 0.0, 0.0]`
`initial_velocity`	`float`	Initial velocity magnitude of the reference node	`0`

NonLinearDynamicMultibody¶

class sharpy.solvers.nonlineardynamicmultibody.NonLinearDynamicMultibody[source]¶

Nonlinear dynamic multibody

Nonlinear dynamic step solver for multibody structures.

The settings that this solver accepts are given by a dictionary, with the following key-value pairs:

Name	Type	Description	Default
`print_info`	`bool`	Print output to screen	`True`
`max_iterations`	`int`	Sets maximum number of iterations	`100`
`num_load_steps`	`int`		`1`
`delta_curved`	`float`		`0.01`
`min_delta`	`float`	Structural solver tolerance	`1e-05`
`newmark_damp`	`float`	Sets the Newmark damping coefficient	`0.0001`
`gravity_on`	`bool`	Flag to include gravitational forces	`False`
`gravity`	`float`	Gravitational acceleration	`9.81`
`gravity_dir`	`list(float)`	Direction in G where gravity applies	`[0.0, 0.0, 1.0]`
`relaxation_factor`	`float`		`0.3`
`dt`	`float`	Time step increment	`0.01`
`num_steps`	`int`		`500`
`time_integrator`	`str`	Method to perform time integration	`NewmarkBeta`
`time_integrator_settings`	`dict`	Settings for the time integrator	`{}`
`write_lm`	`bool`	Write lagrange multipliers	`False`
`relax_factor_lm`	`float`	Relaxation factor for Lagrange Multipliers. 0 no relaxation. 1 full relaxation	`0.0`

assembly_MB_eq_system(MB_beam, MB_tstep, ts, dt, Lambda, Lambda_dot, MBdict)[source]¶

This function generates the matrix and vector associated to the linear system to solve a structural iteration It usses a Newmark-beta scheme for time integration. Being M, C and K the mass, damping and stiffness matrices of the system:

\[MB_Asys = MB_K + MB_C\]

rac{gamma}{eta dt} + rac{1}{eta dt^2} MB_M

Args:
MB_beam (list(Beam)): each entry represents a body MB_tstep (list(StructTimeStepInfo)): each entry represents a body ts (int): Time step number dt(int): time step Lambda (np.ndarray): Lagrange Multipliers array Lambda_dot (np.ndarray): Time derivarive of Lambda MBdict (dict): Dictionary including the multibody information

Returns:
MB_Asys (np.ndarray): Matrix of the systems of equations MB_Q (np.ndarray): Vector of the systems of equations

compute_forces_constraints(MB_beam, MB_tstep, ts, dt, Lambda, Lambda_dot)[source]¶

This function computes the forces generated at Lagrange Constraints

Parameters

MB_beam (list(Beam)) – each entry represents a body
MB_tstep (list(StructTimeStepInfo)) – each entry represents a body
ts (int) – Time step number
dt (float) – Time step increment
Lambda (np.ndarray) – Lagrange Multipliers array
Lambda_dot (np.ndarray) – Time derivarive of Lambda

Warning

This function is underdevelopment and not fully functional

define_sys_size()[source]¶

This function defines the number of degrees of freedom in a multibody systems

Each body contributes with num_dof degrees of freedom and 10 more if the associated local FoR can move or has Lagrange Constraints associated

integrate_position(MB_beam, MB_tstep, dt)[source]¶

This function integrates the position of each local A FoR after the structural iteration has been solved.

It uses a Newmark-beta approximation.

Parameters

MB_beam (list(Beam)) – each entry represents a body
MB_tstep (list(StructTimeStepInfo)) – each entry represents a body
dt (int) – time step

NonLinearDynamicPrescribedStep¶

class sharpy.solvers.nonlineardynamicprescribedstep.NonLinearDynamicPrescribedStep[source]¶

Structural solver used for the dynamic simulation of clamped structures or those subject to a prescribed motion.

This solver provides an interface to the structural library (xbeam) and updates the structural parameters for every k-th step in the FSI iteration.

This solver can be called as part of a standalone structural simulation or as the structural solver of a coupled aeroelastic simulation.

The settings that this solver accepts are given by a dictionary, with the following key-value pairs:

Name	Type	Description	Default
`print_info`	`bool`	Print output to screen	`True`
`max_iterations`	`int`	Sets maximum number of iterations	`100`
`num_load_steps`	`int`		`1`
`delta_curved`	`float`		`0.01`
`min_delta`	`float`	Structural solver tolerance	`1e-05`
`newmark_damp`	`float`	Sets the Newmark damping coefficient	`0.0001`
`gravity_on`	`bool`	Flag to include gravitational forces	`False`
`gravity`	`float`	Gravitational acceleration	`9.81`
`gravity_dir`	`list(float)`	Direction in G where gravity applies	`[0.0, 0.0, 1.0]`
`relaxation_factor`	`float`		`0.3`
`dt`	`float`	Time step increment	`0.01`
`num_steps`	`int`		`500`

NonLinearStatic¶

class sharpy.solvers.nonlinearstatic.NonLinearStatic[source]¶

Structural solver used for the static simulation of free-flying structures.

This solver provides an interface to the structural library (xbeam) and updates the structural parameters for every k-th step of the FSI iteration.

This solver can be called as part of a standalone structural simulation or as the structural solver of a coupled static aeroelastic simulation.

The settings that this solver accepts are given by a dictionary, with the following key-value pairs:

Name	Type	Description	Default
`print_info`	`bool`	Print output to screen	`True`
`max_iterations`	`int`	Sets maximum number of iterations	`100`
`num_load_steps`	`int`		`1`
`delta_curved`	`float`		`0.01`
`min_delta`	`float`	Structural solver tolerance	`1e-05`
`newmark_damp`	`float`	Sets the Newmark damping coefficient	`0.0001`
`gravity_on`	`bool`	Flag to include gravitational forces	`False`
`gravity`	`float`	Gravitational acceleration	`9.81`
`gravity_dir`	`list(float)`	Direction in G where gravity applies	`[0.0, 0.0, 1.0]`
`relaxation_factor`	`float`		`0.3`
`dt`	`float`	Time step increment	`0.01`
`num_steps`	`int`		`500`
`initial_position`	`list(float)`		`numpy.array`

RigidDynamicCoupledStep¶

class sharpy.solvers.rigiddynamiccoupledstep.RigidDynamicCoupledStep[source]¶

Structural solver used for the dynamic simulation of free-flying rigid structures.

This solver provides an interface to the structural library (xbeam) and updates the structural parameters for every k-th step in the FSI iteration.

This solver can be called as part of a standalone structural simulation or as the structural solver of a coupled aeroelastic simulation.

The settings that this solver accepts are given by a dictionary, with the following key-value pairs:

Name	Type	Description	Default
`print_info`	`bool`	Print output to screen	`True`
`max_iterations`	`int`	Sets maximum number of iterations	`100`
`num_load_steps`	`int`		`1`
`delta_curved`	`float`		`0.01`
`min_delta`	`float`	Structural solver tolerance	`1e-05`
`newmark_damp`	`float`	Sets the Newmark damping coefficient	`0.0001`
`gravity_on`	`bool`	Flag to include gravitational forces	`False`
`gravity`	`float`	Gravitational acceleration	`9.81`
`gravity_dir`	`list(float)`	Direction in G where gravity applies	`[0.0, 0.0, 1.0]`
`relaxation_factor`	`float`		`0.3`
`dt`	`float`	Time step increment	`0.01`
`num_steps`	`int`		`500`
`balancing`	`bool`		`False`

RigidDynamicPrescribedStep¶

class sharpy.solvers.rigiddynamicprescribedstep.RigidDynamicPrescribedStep[source]¶

@modified Alfonso del Carre

The settings that this solver accepts are given by a dictionary, with the following key-value pairs:

Name

Type

Description

Default

print_info

bool

Print output to screen

True

max_iterations

int

Sets maximum number of iterations

100

num_load_steps

int

1

delta_curved

float

0.01

min_delta

float

Structural solver tolerance

1e-05

newmark_damp

float

Sets the Newmark damping coefficient

0.0001

gravity_on

bool

Flag to include gravitational forces

False

gravity

float

Gravitational acceleration

9.81

gravity_dir

list(float)

Direction in G where gravity applies

[0.0, 0.0, 1.0]

relaxation_factor

float

0.3

dt

float

Time step of simulation

0.01

num_steps

int

Number of timesteps to be run

500

Time_integrator Solvers¶

NewmarkBeta¶

GeneralisedAlpha¶

Post-Processing¶

AeroForcesCalculator¶

AerogridPlot¶

AsymptoticStability¶

BeamLoads¶

BeamPlot¶

class sharpy.postproc.beamplot.BeamPlot[source]¶

Plots beam to Paraview format

The settings that this solver accepts are given by a dictionary, with the following key-value pairs:

Name	Type	Description	Default
`include_rbm`	`bool`	Include frame of reference rigid body motion	`True`
`include_FoR`	`bool`	Include frame of reference variables	`False`
`include_applied_forces`	`bool`	Write beam applied forces	`True`
`include_applied_moments`	`bool`	Write beam applied moments	`True`
`name_prefix`	`str`	Name prefix for files
`output_rbm`	`bool`	Write `csv` file with rigid body motion data	`True`

Cleanup¶

FrequencyResponse¶

LiftDistribution¶

class sharpy.postproc.liftdistribution.LiftDistribution[source]¶

Calculates and exports the lift distribution on lifting surfaces

The settings that this solver accepts are given by a dictionary, with the following key-value pairs:

Name	Type	Description	Default
`text_file_name`	`str`	Text file name	`lift_distribution.csv`
`coefficients`	`bool`	Calculate aerodynamic lift coefficients	`True`
`rho`	`float`	Reference freestream density [kg/m³]	`1.225`

PickleData¶

PlotFlowField¶

class sharpy.postproc.plotflowfield.PlotFlowField[source]¶

Plots the flow field in Paraview and computes the velocity at a set of points in a grid.

The settings that this solver accepts are given by a dictionary, with the following key-value pairs:

Name	Type	Description	Default	Options
`postproc_grid_generator`	`str`	Generator used to create grid and plot flow field	`GridBox`	`GridBox`
`postproc_grid_input`	`dict`	Dictionary containing settings for `postproc_grid_generator`.	`{}`
`velocity_field_generator`	`str`	Chosen velocity field generator	`SteadyVelocityField`
`velocity_field_input`	`dict`	Dictionary containing settings for the selected `velocity_field_generator`.	`{}`
`dt`	`float`	Time step.	`0.1`
`include_external`	`bool`	Include external velocities.	`True`
`include_induced`	`bool`	Include induced velocities.	`True`
`stride`	`int`	Number of time steps between plots.	`1`
`num_cores`	`int`	Number of cores to use.	`1`
`vortex_radius`	`float`	Distance below which inductions are not computed.	`sharpy.utils.constants.vortex_radius_def`

SaveData¶

SaveParametricCase¶

StallCheck¶

UDPout¶

WriteVariablesTime¶

SHARPy Source Code¶

The core SHARPy documentation is found herein.

Note

The docs are still a work in progress and therefore, most functions/classes with which there is not much user interaction are not fully documented. We would appreciate any help by means of you contributing to our growing documentation!

If you feel that a function/class is not well documented and, hence, you cannot use it, feel free to raise an issue so that we can improve it.

Aerodynamic Packages¶

Models¶

Aerogrid¶

Aerogrid contains all the necessary routines to generate an aerodynamic grid based on the input dictionaries.

AeroTimeStepInfo¶

class sharpy.aero.models.aerogrid.AeroTimeStepInfo(dimensions, dimensions_star)[source]¶

Aerodynamic Time step class.

Contains the relevant aerodynamic attributes for a single time step. All variables should be expressed in G FoR unless otherwise stated.

ct_dimensions¶: Pointer to dimensions to interface the C++ library uvlmlib`

ct_dimensions_star¶: Pointer to dimensions_star to interface the C++ library uvlmlib`

dimensions¶

Matrix defining the dimensions of the vortex grid on solid surfaces [num_surf x chordwise panels x spanwise panels]

Type: np.ndarray

dimensions_star¶

Matrix defining the dimensions of the vortex grid on wakes [num_surf x streamwise panels x spanwise panels]

Type: np.ndarray

n_surf¶

Number of aerodynamic surfaces on solid bodies. Each aerodynamic surface on solid bodies will have an associted wake.

Type: int

zeta¶

Location of solid grid vertices [n_surf][3 x (chordwise nodes + 1) x (spanwise nodes + 1)]

Type: list(np.ndarray

zeta_dot¶

Time derivative of zeta

Type: list(np.ndarray)

normals¶

Normal direction to panels at the panel center [n_surf][3 x chordwise nodes x spanwise nodes]

Type: list(np.ndarray)

forces¶

Forces not associated to time derivatives on grid vertices [n_surf][3 x (chordwise nodes + 1) x (spanwise nodes + 1)]

Type: list(np.ndarray)

dynamic_forces¶

Forces associated to time derivatives on grid vertices [n_surf][3 x (chordwise nodes + 1) x (spanwise nodes + 1)]

Type: list(np.ndarray)

zeta_star¶

Location of wake grid vertices [n_surf][3 x (streamwise nodes + 1) x (spanwise nodes + 1)]

Type: list(np.ndarray

u_ext¶

Background flow velocity on solid grid nodes [n_surf][3 x (chordwise nodes + 1) x (spanwise nodes + 1)]

Type: list(np.ndarray)

u_ext_star¶

Background flow velocity on wake grid nodes [n_surf][3 x (streamwise nodes + 1) x (spanwise nodes + 1)]

Type: list(np.ndarray)

gamma¶

Circulation associated to solid panels [n_surf][3 x chordwise nodes x spanwise nodes]

Type: list(np.ndarray)

gamma_star¶

Circulation associated to wake panels [n_surf][3 x streamwise nodes x spanwise nodes]

Type: list(np.ndarray)

gamma_dot¶

Time derivative of gamma

Type: list(np.ndarray)

inertial_steady_forces¶

Total aerodynamic steady forces in G FoR [n_surf x 6]

Type: list(np.ndarray)

body_steady_forces¶

Total aerodynamic steady forces in A FoR [n_surf x 6]

Type: list(np.ndarray)

inertial_unsteady_forces¶

Total aerodynamic unsteady forces in G FoR [n_surf x 6]

Type: list(np.ndarray)

body_unsteady_forces¶

Total aerodynamic unsteady forces in A FoR [n_surf x 6]

Type: list(np.ndarray)

postproc_cell¶

Variables associated to cells to be postprocessed

Type: dict

postproc_node¶

Variables associated to nodes to be postprocessed

Type: dict

in_global_AFoR¶

True if the variables are stored in the global A FoR. False if they are stored in the local A FoR of each body. Always True for single-body simulations. Currently not used.

Type: bool

control_surface_deflection¶

Deflection of the control surfaces, in rad and if fitted.

Type: np.ndarray

Parameters

dimensions (np.ndarray) – Matrix defining the dimensions of the vortex grid on solid surfaces [num_surf x chordwise panels x spanwise panels]
dimensions_star (np.ndarray) – Matrix defining the dimensions of the vortex grid on wakes [num_surf x streamwise panels x spanwise panels]

copy()[source]¶: Returns a copy of a deepcopy of a AeroTimeStepInfo

generate_ctypes_pointers()[source]¶: Generates the pointers to aerodynamic variables used to interface the C++ library uvlmlib

remove_ctypes_pointers()[source]¶: Removes the pointers to aerodynamic variables used to interface the C++ library uvlmlib

Aerogrid¶

class sharpy.aero.models.aerogrid.Aerogrid[source]¶

Aerogrid is the main object containing information of the grid of panels

It is created by the solver sharpy.solvers.aerogridloader.AerogridLoader

static compute_gamma_dot(dt, tstep, previous_tsteps)[source]¶

Computes the temporal derivative of circulation (gamma) using finite differences.

It will use a first order approximation for the first evaluation (when len(previous_tsteps) == 1), and then second order ones.

\[\left.\frac{d\Gamma}{dt}\right|^n \approx \lim_{\Delta t \rightarrow 0}\frac{\Gamma^n-\Gamma^{n-1}}{\Delta t}\]

For the second time step and onwards, the following second order approximation is used:

\[\left.\frac{d\Gamma}{dt}\right|^n \approx \lim_{\Delta t \rightarrow 0}\frac{3\Gamma^n -4\Gamma^{n-1}+\Gamma^{n-2}}{2\Delta t}\]

Parameters

dt (float) – delta time for the finite differences
tstep (AeroTimeStepInfo) – tstep at time n (current)
previous_tsteps (list(AeroTimeStepInfo)) – previous tstep structure in order: [n-N,..., n-2, n-1]

Returns

first derivative of circulation with respect to time

Return type

float

See also

class sharpy.utils.datastructures.AeroTimeStepInfo¶

generate_strip¶

Returns a strip of panels in A frame of reference, it has to be then rotated to simulate angles of attack, etc

Aerodynamic Utilities¶

Polar¶

class sharpy.aero.utils.airfoilpolars.Polar[source]¶

Airfoil polar object

initialise(table)[source]¶

Initialise polar

Parameters: table (np.ndarray) – 4-column array containing aoa (rad), cl, cd and cm

Force Mapping Utilities¶

aero2struct_force_mapping¶

Maps the aerodynamic forces at the lattice to the structural nodes

The aerodynamic forces from the UVLM are always in the inertial G frame of reference and have to be transformed to the body or local B frame of reference in which the structural forces are defined.

Since the structural nodes and aerodynamic panels are coincident in a spanwise direction, the aerodynamic forces that correspond to a structural node are the summation of the M+1 forces defined at the lattice at that spanwise location.

\[\begin{split}\mathbf{f}_{struct}^B &= \sum\limits_{i=0}^{m+1}C^{BG}\mathbf{f}_{i,aero}^G \\ \mathbf{m}_{struct}^B &= \sum\limits_{i=0}^{m+1}C^{BG}(\mathbf{m}_{i,aero}^G + \tilde{\boldsymbol{\zeta}}^G\mathbf{f}_{i, aero}^G)\end{split}\]

where $\tilde{\boldsymbol{\zeta}}^G$ is the skew-symmetric matrix of the vector between the lattice grid vertex and the structural node.

param aero_forces: Aerodynamic forces from the UVLM in inertial frame of reference
type aero_forces: list
param struct2aero_mapping: Structural to aerodynamic node mapping
type struct2aero_mapping: dict
param zeta: Aerodynamic grid coordinates
type zeta: list
param pos_def: Vector of structural node displacements
type pos_def: np.ndarray
param psi_def: Vector of structural node rotations (CRVs)
type psi_def: np.ndarray
param master: Unused
param conn: Connectivities matrix
type conn: np.ndarray
param cag: Transformation matrix between inertial and body-attached reference A
type cag: np.ndarray
param aero_dict: Dictionary containing the grid’s information.
type aero_dict: dict
returns: structural forces in an n_node x 6 vector
rtype: np.ndarray

total_forces_moments¶

Performs a summation of the forces and moments expressed at the structural nodes in the A frame of reference.

Note

If you need to transform forces and moments at the nodes from B to A, use the nodal_b_for_2_a_for() function.

param forces_nodes_a: n_node x 6 vector of forces and moments at the nodes in A
type forces_nodes_a: np.array
param pos_def: n_node x 3 vector of nodal positions in A
type pos_def: np.array
param ref_pos: Location in A about which to compute moments. Defaults to [0, 0, 0]
type ref_pos: np.array (optional)
returns: Vector of length 6 containing the total forces and moments expressed in A at the desired location.
rtype: np.array

Controllers¶

ControlSurfacePidController¶

class sharpy.controllers.controlsurfacepidcontroller.ControlSurfacePidController[source]¶

The settings that this solver accepts are given by a dictionary, with the following key-value pairs:

Name	Type	Description	Default
`time_history_input_file`	`str`	Route and file name of the time history of desired state	`None`
`P`	`float`	Proportional gain of the controller	`None`
`I`	`float`	Integral gain of the controller	`0.0`
`D`	`float`	Differential gain of the controller	`0.0`
`input_type`	`str`	Quantity used to define the reference state. Supported: pitch	`None`
`dt`	`float`	Time step of the simulation	`None`
`controlled_surfaces`	`list(int)`	Control surface indices to be actuated by this controller	`None`
`controlled_surfaces_coeff`	`list(float)`	Control surface deflection coefficients. For example, for antisymmetric deflections => [1, -1].	`[1.0]`
`write_controller_log`	`bool`	Write a time history of input, required input, and control	`True`
`controller_log_route`	`str`	Directory where the log will be stored	`./output/`

control(data, controlled_state)[source]¶

Main routine of the controller. Input is data (the self.data in the solver), and currrent_state which is a dictionary with [‘structural’, ‘aero’] time steps for the current iteration.

Parameters

data – problem data containing all the information.
controlled_state – dict with two vars: structural and aero containing the timestep_info that will be returned with the control variables.

Returns

A dict with structural and aero time steps and control input included.

TakeOffTrajectoryController¶

class sharpy.controllers.takeofftrajectorycontroller.TakeOffTrajectoryController[source]¶

The settings that this solver accepts are given by a dictionary, with the following key-value pairs:

Name	Type	Description	Default
`trajectory_input_file`	`str`	Route and file name of the trajectory file given as a csv with columns: time, x, y, z	`None`
`dt`	`float`	Time step of the simulation	`None`
`trajectory_method`	`str`	Trajectory controller method. For now, “lagrange” is the supported option	`lagrange`
`controlled_constraint`	`str`	Name of the controlled constraint in the multibody context Usually, it is something like constraint_00.	`None`
`controller_log_route`	`str`	Directory where the log will be stored	`./output/`
`write_controller_log`	`bool`	Controls if the log from the controller is written or not.	`True`
`free_trajectory_structural_solver`	`str`	If different than and empty string, the structural solver will be changed after the end of the trajectory has been reached
`free_trajectory_structural_substeps`	`int`	Controls the structural solver structural substeps once the end of the trajectory has been reached	`0`
`initial_ramp_length_structural_substeps`	`int`	Controls the number of timesteps that are used to increase the structural substeps from 0	`10`

control(data, controlled_state)[source]¶

Main routine of the controller. Input is data (the self.data in the solver), and currrent_state which is a dictionary with [‘structural’, ‘aero’] time steps for the current iteration.

Parameters

data – problem data containing all the information.
controlled_state – dict with two vars: structural and aero containing the timestep_info that will be returned with the control variables.

Returns

A dict with structural and aero time steps and control input included.

process_trajectory(dxdt=True)[source]¶: See https://docs.scipy.org/doc/scipy-0.15.1/reference/generated/scipy.interpolate.UnivariateSpline.html

Generators¶

Velocity field generators prescribe the flow conditions for your problem. For instance, you can have an aircraft at a prescribed fixed location in a velocity field towards the aircraft. Alternatively, you can have a free moving aircraft in a static velocity field.

Dynamic Control Surface generators enable the user to prescribe a certain control surface deflection in time.

BumpVelocityField¶

DynamicControlSurface¶

class sharpy.generators.dynamiccontrolsurface.DynamicControlSurface[source]¶

Dynamic Control Surface deflection Generator

The object generates a deflection in radians based on the time series given as a single vector in the input data. A first order finite-differences scheme is used to calculate the deflection rate based on the provided time step increment.

To call this generator, the generator_id = DynamicControlSurface key shall be used for the setting control_surface_deflection in the AerogridLoader solver.

One instance of this generator will be created for each control surface, thus, a group of settings should be defined for each control surface (cs0_settings, cs1_settings … in the example below). All of these groups of settings should be collected as values in a dictionary which keys are the associated control surface number in string format. This dictionary should be parsed to the variable

control_surface_deflection_generator_settings in AerogridLoader. This is shown better in the example below:

Examples:

Attributes:
deflection (np.array): Array of deflection of the control surface deflection_dot (np.array): Array of the time derivative of the cs deflection. Calculated using 1st order finite differences.

This generator takes the following inputs:

Name

Type

Description

Default

dt

float

Time step increment

None

deflection_file

str

Path to the file with the deflection information

None

FloatingForces¶

change_of_to_sharpy¶

compute_equiv_hd_added_mass¶

compute_jacobian¶

compute_xf_zf¶

jonswap_spectrum¶

matrix_from_rf¶

noise_freq_1s¶

quasisteady_mooring¶

rename_terms¶

response_freq_dep_matrix¶

rfval¶

time_wave_forces¶

wave_radiation_damping¶

GridBox¶

Gust Velocity Field Generators¶

These generators are used to create a gust velocity field. GustVelocityField is the main class that should be parsed as the velocity_field_input to the desired aerodynamic solver.

The remaining classes are the specific gust profiles and parsed as gust_shape.

Examples:

The typical input to the aerodynamic solver settings would therefore read similar to:

>>> aero_settings = {'<some_aero_settings>': '<some_aero_settings>',
>>>                  'velocity_field_generator': 'GustVelocityField',
>>>                  'velocity_field_input': {'u_inf': 1,
>>>                                           'gust_shape': '<desired_gust>',
>>>                                           'gust_parameters': '<gust_settings>'}}

DARPA¶

GustVelocityField¶

continuous_sin¶

lateral_one_minus_cos¶

one_minus_cos¶

span_sine¶

time_varying¶

time_varying_global¶

HelicoidalWake¶

class sharpy.generators.helicoidalwake.HelicoidalWake[source]¶

Helicoidal wake shape generator

HelicoidalWake class inherited from BaseGenerator

The object creates a helicoidal wake shedding from the trailing edge based on the time step dt, the incoming velocity magnitude u_inf, direction u_inf_direction, the rotation velocity rotation_velocity and the shear parameters

The settings that this solver accepts are given by a dictionary, with the following key-value pairs:

Name	Type	Description	Default
`u_inf`	`float`	Free stream velocity magnitude	`None`
`u_inf_direction`	`list(float)`	`x`, `y` and `z` relative components of the free stream velocity	`None`
`dt`	`float`	Time step	`None`
`dphi1`	`float`	Size of the first wake panel in radians	`-1.0`
`ndphi1`	`int`	Number of panels with size `dphi1`	`1`
`r`	`float`	Growth rate after `ndphi1` panels	`1.0`
`dphimax`	`float`	Maximum panel size in radians	`-1.0`
`shear_direction`	`list(float)`	`x`, `y` and `z` relative components of the direction along which shear applies	`numpy.array`
`shear_exp`	`float`	Exponent of the shear law	`0.0`
`h_ref`	`float`	Reference height at which `u_inf` is defined	`1.0`
`h_corr`	`float`	Height to correct the shear law	`1.0`
`rotation_velocity`	`list(float)`	Rotation velocity	`None`

ChangeLumpedMass¶

ChangedVariable¶

LumpedMassControl¶

ModifyStructure¶

EfficiencyCorrection¶

class sharpy.generators.polaraeroforces.EfficiencyCorrection[source]¶

The efficiency and constant terms are introduced by means of the array airfoil_efficiency in the aero.h5

\[\mathbf{f}_{struct}^B &=\]

arepsilon^f_0 mathbf{f}_{i,struct}^B + arepsilon^f_1

mathbf{m}_{struct}^B &=

arepsilon^m_0 mathbf{m}_{i,struct}^B + arepsilon^m_1

Notice that the moment correction is applied on top of the force correction. As a consequence, the aerodynamic moments generated by the forces on the vertices are corrected sequentially by both efficiencies.

See Also:
The SHARPy case files documentation for a detailed overview on how to include the airfoil efficiencies.

Returns:
np.ndarray: corresponding aerodynamic force at the structural node from the force and moment at a grid vertex

generate(**params)[source]¶

Keyword Arguments

aero_kstep (sharpy.utils.datastructures.AeroTimeStepInfo) – Current aerodynamic substep
structural_kstep (sharpy.utils.datastructures.StructTimeStepInfo) – Current structural substep
struct_forces (np.array) – Array with the aerodynamic forces mapped on the structure in the B frame of reference

Returns

New corrected structural forces

Return type

np.array

PolarCorrection¶

class sharpy.generators.polaraeroforces.PolarCorrection[source]¶

This generator corrects the aerodynamic forces from UVLM based on the airfoil polars provided by the user in the aero.h5 file. Polars are entered for each airfoil, in a table comprising AoA (rad), CL, CD, CM.

This generator_id = 'PolarCorrection' and can be used in the coupled solvers through the correct_forces_method setting as:

These are the steps needed to correct the forces:

The force coming from UVLM is divided into induced drag (parallel to the incoming flow velocity) and lift

(the remaining force).

If cd_from_cl == 'on'.

The viscous drag and pitching moment are found at the computed lift coefficient. Then forces and moments are updated

Else, the angle of attack is computed:

The angle of attack is computed based on that lift force and the angle of zero lift computed from the airfoil polar and assuming the potential flow lift curve slope of $2 \pi$

The drag force is computed based on the angle of attack and the polars provided by the user

If correct_lift == 'on', the lift coefficient is also corrected with the polar data. Else, only the UVLM results are used.

The pitching moment is added in a similar manner as the viscous drag. However, if moment_from_polar == 'on' and correct_lift == 'on', the total moment (the one used for the FSI) is computed just from polar data, overriding any moment computed in SHARPy. That is, the moment will include the polar pitching moment, and moments due to lift and drag computed from the polar data.

This generator takes in the following settings.

Name	Type	Description	Default
`correct_lift`	`bool`	Correct lift according to the polars	`False`
`cd_from_cl`	`bool`	Interpolate the C_D for the given C_L, as opposed to getting the C_D from the section AoA.	`False`
`moment_from_polar`	`bool`	If `correct_lift` is selected, it will compute the pitching moment simply from polar derived data, i.e. the polars Cm and the momentsarising from the lift and drag (derived from the polar) contribution. Else, it will add the polar Cm to the moment already computed by SHARPy.	`False`

generate(**params)[source]¶

Keyword Arguments

aero_kstep (sharpy.utils.datastructures.AeroTimeStepInfo) – Current aerodynamic substep
structural_kstep (sharpy.utils.datastructures.StructTimeStepInfo) – Current structural substep
struct_forces (np.array) – Array with the aerodynamic forces mapped on the structure in the B frame of reference

Returns

New corrected structural forces

Return type

np.array

get_aoacl0_from_camber¶

This section provies the angle of attach of zero lift for a thin airfoil which camber line is defined by ‘x’ and ‘y’ coordinates

Check Theory of wing sections. Abbott. pg 69

local_stability_axes¶

Rotates the body axes onto stability axes. This rotation is equivalent to the projection of a vector in S onto B.

The stability axes are defined as:

x_s: parallel to the free stream

z_s: perpendicular to the free stream and part of the plane formed by the local chord and the vertical body axis z_b.

y_s: completes the set

param dir_urel: Unit vector in the direction of the free stream velocity expressed in B frame.
type dir_urel: np.array
param dir_chord: Unit vector in the direction of the local chord expressed in B frame.
type dir_chord: np.array
returns: Rotation matrix from B to S, equivalent to the projection matrix $C^{BS}$ that projects a vector from S onto B.
rtype: np.array

magnitude_and_direction_of_relative_velocity¶

Calculates the magnitude and direction of the relative velocity u_rel at a local section of the wing.

\[u_{rel, i}^G = \bar{U}_{\infty, i}^G - C^{GA}(\chi)(\dot{\eta}_i^A + v^A + \tilde{\omega}^A\eta_i^A\]

where $\bar{U}_{\infty, i}^G$ is the average external velocity across all aerodynamic nodes at the relevant cross section.

param displacement: Unit vector in the direction of the free stream velocity expressed in A frame.
type displacement: np.array
param displacement_vel: Unit vector in the direction of the local chord expressed in A frame.
type displacement_vel: np.array
param for_vel: A frame of reference (FoR) velocity. Expressed in A FoR
type for_vel: np.array
param cga: Rotation vector from FoR G to FoR A
type cga: np.array
param uext: Background flow velocity on solid grid nodes
type uext: np.array
returns: u_rel, dir_u_rel expressed in the inertial, G frame.
rtype: tuple

span_chord¶

Retrieve the local span and local chord

param i_node_surf: Node index in aerodynamic surface
type i_node_surf: int
param zeta: Aerodynamic surface coordinates (3 x n_chord x m_span)
type zeta: np.array
returns: dir_span, span, dir_chord, chord
rtype: tuple

ShearVelocityField¶

class sharpy.generators.shearvelocityfield.ShearVelocityField[source]¶

Shear Velocity Field Generator

ShearVelocityField class inherited from BaseGenerator

The object creates a steady velocity field with shear

\[\hat{u} = \hat{u}\_\infty \left( \frac{h - h\_\mathrm{corr}}{h\_\mathrm{ref}} \right)^{\mathrm{shear}\_\mathrm{exp}}\]

\[h = \zeta \cdot \mathrm{shear}\_\mathrm{direction}\]

The settings that this solver accepts are given by a dictionary, with the following key-value pairs:

Name	Type	Description	Default
`u_inf`	`float`	Free stream velocity magnitude	`None`
`u_inf_direction`	`list(float)`	`x`, `y` and `z` relative components of the free stream velocity	`numpy.array`
`shear_direction`	`list(float)`	`x`, `y` and `z` relative components of the direction along which shear applies	`numpy.array`
`shear_exp`	`float`	Exponent of the shear law	`0.0`
`h_ref`	`float`	Reference height at which `u_inf` is defined	`1.0`
`h_corr`	`float`	Height to correct shear law	`0.0`

SteadyVelocityField¶

StraightWake¶

class sharpy.generators.straightwake.StraightWake[source]¶

Straight wake shape generator

This generator creates a straight wake shedding from the trailing edge based on the time step dt, the incoming velocity magnitude u_inf and direction u_inf_direction. It is to be used as wake_generator in

AerogridLoader.

A wake where panels grow downstream is supported by using the settings dx1, ndx1, r and dxmax as described below. Note that the wake will always have m_star panels, as specified in AerogridLoader, thus these settings will modify the effective length of the wake. Once the maximum size of panel dxmax is achieved, all panels are size dxmax thereinafter until m_star panels are created.

The settings that this solver accepts are given by a dictionary, with the following key-value pairs:

Name	Type	Description	Default
`u_inf`	`float`	Free stream velocity magnitude	`1.0`
`u_inf_direction`	`list(float)`	`x`, `y` and `z` relative components of the free stream velocity	`numpy.array`
`dt`	`float`	Time step	`0.1`
`dx1`	`float`	Size of the first wake panel	`-1.0`
`ndx1`	`int`	Number of panels with size `dx1`	`1`
`r`	`float`	Growth rate after `ndx1` panels	`1.0`
`dxmax`	`float`	Maximum panel size	`-1.0`

TrajectoryGenerator¶

TurbVelocityField¶

class sharpy.generators.turbvelocityfield.TurbVelocityField[source]¶

Turbulent Velocity Field Generator

TurbVelocitityField is a class inherited from BaseGenerator

The TurbVelocitityField class generates a velocity field based on the input from an [XDMF](http://www.xdmf.org) file. It supports time-dependant fields as well as frozen turbulence.

To call this generator, the generator_id = TurbVelocityField shall be used. This is parsed as the value for the velocity_field_generator key in the desired aerodynamic solver’s settings.

Supported files:

field_id.xdmf: Steady or Unsteady XDMF file

This generator also performs time interpolation between two different time steps. For now, only linear interpolation is possible.

Space interpolation is done through scipy.interpolate trilinear interpolation. However, turbulent fields are read directly from the binary file and not copied into memory. This is performed using np.memmap. The overhead of this procedure is ~18% for the interpolation stage, however, initially reading the binary velocity field (which will be much more common with time-domain simulations) is faster by a factor of 1e4. Also, memory savings are quite substantial: from 6Gb for a typical field to a handful of megabytes for the whole program.

Parameters: in_dict (dict) – Input data in the form of dictionary. See acceptable entries below:

Attributes:

See also

class sharpy.utils.generator_interface.BaseGenerator¶

The settings that this solver accepts are given by a dictionary, with the following key-value pairs:

Name	Type	Description	Default
`print_info`	`bool`	Output solver-specific information in runtime.	`True`
`turbulent_field`	`str`	XDMF file path of the velocity field	`None`
`offset`	`list(float)`	Spatial offset in the 3 dimensions	`numpy.zeros`
`centre_y`	`bool`	Flat for changing the domain to [`-y_max/2`, `y_max/2`]	`True`
`periodicity`	`str`	Axes in which periodicity is enforced	`xy`
`frozen`	`bool`	If `True`, the turbulent field will not be updated in time	`True`
`store_field`	`bool`	If `True`, the xdmf snapshots are stored in memory. Only two at a time for the linear interpolation	`False`

read_btl(in_file)[source]¶: Legacy function, not using the custom format based on HDF5 anymore.

read_grid(i_grid, i_cache=0)[source]¶: This function returns an interpolator list of size 3 made of scipy.interpolate.RegularGridInterpolator objects.

read_xdmf(in_file)[source]¶

Reads the xml file <case_name>.xdmf. Writes the self.grid_data data structure with all the information necessary.

Note: this function does not load any turbulence data (such as ux000, …), it only reads the header information contained in the xdmf file.

TurbVelocityFieldBts¶

UDP Input/Output¶

This package contains the routines for the SHARPy input and output via UDP.

The main interface is performed through the NetworkLoader

SetOfVariables¶

class sharpy.io.inout_variables.SetOfVariables[source]¶

Iterable class containing the input and output variables

variables¶

List of Variable

Type: list(Variable)

encode()[source]¶

Encode output variables in binary format with little-endian byte ordering.

The signal consists of a 5-byte header RREF0 followed by 8 bytes per variable. Of those 8 bytes allocated to each variable, the first 4 are the integer value of the variable index and the last 4 are the single precision float value.

Returns: Encoded message of length 5 + num_var * 8.
Return type: bytes

get_value(data, timestep_index=- 1)[source]¶

Gets the value from the data structure for output variables

Parameters

data (sharpy.presharpy.PreSharpy) – the standard SHARPy class
timestep_index (int (optional)) – Integer representing the time step value. Defaults to -1 i.e. the last one available.

set_value(values)[source]¶

Sets the values of the input variables.

Parameters: values (list(tuple)) – List of tuples containing the index and value of the respective input variables.

InNetwork¶

class sharpy.io.network_interface.InNetwork[source]¶

Input Network socket settings

Note

If sending/receiving data across the net or LAN, make sure that your firewall has the desired ports open, otherwise the signals will not make it through.

The settings that this solver accepts are given by a dictionary, with the following key-value pairs:

Name	Type	Description	Default
`address`	`str`	Own network address.	`127.0.0.1`
`port`	`int`	Own port for input network	`65001`

Network¶

class sharpy.io.network_interface.Network(host=None, port=None)[source]¶

Network Adapter

Contains the basic methods. See InNetwork and OutNetwork for specific settings pertaining to the input and output sockets.

The settings that this solver accepts are given by a dictionary, with the following key-value pairs:

Name	Type	Description	Default
`address`	`str`	Own network address.	`127.0.0.1`
`port`	`int`	Own port.	`65000`

set_client_list(list_of_clients)[source]¶

Set a client list for network.

Parameters: list_of_clients (list) – List of tuples containing (HOST, PORT), where HOST is a string and port and integer.

set_selector_events_mask(mode)[source]¶: Set selector to listen for events: mode is ‘r’, ‘w’, or ‘rw’.

NetworkLoader¶

class sharpy.io.network_interface.NetworkLoader[source]¶

SHARPy UDP data input and output interface.

The settings of this interface are to be used as the dictionary to the setting network_setting in the DynamicCoupled solver, which is the only one that is currently supported.

This interface allows for SHARPy to receive and send simulation data over the network using an UDP protocol.

The setting variables_filename is a filename to a YAML file that contains a list of the input or output variables. The example below shows an acceptable input

---
- name: 'control_surface_deflection' # variable name. those in the timestep_info are supported
  var_type: 'control_surface'
  inout: 'in'  # either `in`, `out` or `inout`
  position: 0  # control surface index
- name: 'pos'  # variable name
  var_type: 'node'  # type of variable. In this case a node variable. Others: `panel`, `control_surface`
  inout: 'out'
  position: 5  # node number
  index: 2  # vector index, in this case a 3d vector where the desired index is number 2
- name: 'gamma'
  inout: 'out'
  position: [0, 1, 2] # [i_surf, i_chordwise, i_spanwise]
  var_type: 'panel'
- name: 'psi'  # CRV
  inout: 'out'
  var_type: 'node'
  position: 3  # node id
  index: 2  # dimension index
...

All variables in the aero and structural timestep info classes datastructures are supported, with the addition of dt for the time increment and nt for the current time step number.

Note

If using a control surface input, make sure this control surface is given control_surface_type = 2 in the the case .aero.h5 file. Otherwise, the control surface will not move!

The relevant settings for the input and output sockets can be found in InNetwork and OutNetwork, respectively.

If the setting send_output_to_all_clients is True, then the clients from which the input signal is received will also be added to the destination client address book.

The input and output messages follow the example set by X-Plane RREF0 protocol. Thus, a message consists of a 5-byte header containing RREF0 followed by 8-bytes per variable, where the first 4-bytes correspond to the variable number (as ordered in the YAML file) as an integer and the latter 4-bytes correspond to the value of the variable in single precision float. The byte ordering is specified by the user.

A specific network log is created to detail the ins and outs of the communication protocol. The level of messages that are shown can be set in the settings.

Note

The SHARPy input and output sockets do not time out.

Note

The first time step in a simulation with UDP inputs takes particularly long. Make sure your client has a sufficient time out time to avoid issues. After the first time step, the UDP should not delay the simulation.

Warning

There is a limitation, for the moment, on just one control surface being supported for UDP input.

The NetworkLoader takes the following settings:

Name	Type	Description	Default
`variables_filename`	`str`	Path to YAML file containing input/output variables	`None`
`byte_ordering`	`str`	Desired endianness byte ordering	`little`
`input_network_settings`	`dict`	Settings for the input network.:class:~sharpy.io.network_interface.InNetwork.	`{}`
`output_network_settings`	`dict`	Settings for the output network `OutNetwork`.	`{}`
`send_output_to_all_clients`	`bool`	Send output to all clients, including those from where the input is received.	`False`
`received_data_filename`	`str`	If not empty, writes received input data to the specified file.
`log_name`	`str`	Network log file name	`./network_output.log`
`console_log_level`	`str`	Minimum logging level in console.	`info`
`file_log_level`	`str`	Minimum logging level in log file.	`debug`

OutNetwork¶

class sharpy.io.network_interface.OutNetwork(host=None, port=None)[source]¶

Output network socket settings

If send_on_demand is True, SHARPy will only output data when it receives a request for it. The request message can be any message under 1024 bytes. SHARPy will reply to the socket that sent the request with the latest time step information. Otherwise, it will send data at the end of each time step to the specified destination clients.

If the NetworkLoader setting send_output_to_all_clients is True, then the clients from which the input signal is received will also be added to the destination client address book.

Note

If sending/receiving data across the net or LAN, make sure that your firewall has the desired ports open, otherwise the signals will not make it through.

The settings that this solver accepts are given by a dictionary, with the following key-value pairs:

Name	Type	Description	Default
`address`	`str`	Own network address.	`127.0.0.1`
`port`	`int`	Own port for output network	`65000`
`send_on_demand`	`bool`	Waits for a signal demanding the output data. Else, sends to destination buffer	`True`
`destination_address`	`list(str)`	List of addresses to send output data. If `send_on_demand` is `False` this is a required setting.	`[]`
`destination_ports`	`list(int)`	List of ports number for the destination addresses.	`[]`

Linear SHARPy¶

The code included herein enables the assembly of linearised state-space systems based on the previous solution of a nonlinear problem that will be used as linearisation reference.

The code is structured in the following way:

Assembler: different state-spaces to assemble, from only structural/aerodynamic to fully coupled aeroelastic

Src: source code required for the linearisation and utilities for the manipulation of state-space elements

References:

Maraniello, S. , Palacios, R.. State-Space Realizations and Internal Balancing in Potential-Flow Aerodynamics with Arbitrary Kinematics. AIAA Journal, Vol. 57, No.6, June 2019

Assembler¶

Control surface deflector for linear systems¶

Control surface deflector for linear systems

LinControlSurfaceDeflector¶

class sharpy.linear.assembler.lincontrolsurfacedeflector.LinControlSurfaceDeflector[source]¶

Subsystem that deflects control surfaces for use with linear state space systems

The current version supports only deflections. Future work will include standalone state-space systems to model physical actuators.

generate()[source]¶

Generates a matrix mapping a linear control surface deflection onto the aerodynamic grid.

This generates two matrices:

Kzeta_delta maps the deflection angle onto displacements. It has as many columns as independent control surfaces.

Kdzeta_ddelta maps the deflection rate onto grid velocities. Again, it has as many columns as independent control surfaces.

Returns: Tuple containing Kzeta_delta and Kdzeta_ddelta.
Return type: tuple

der_R_arbitrary_axis_times_v¶

Linearised rotation vector of the vector v by angle theta about an arbitrary axis u.

The rotation of a vector $\mathbf{v}$ about the axis $\mathbf{u}$ by an angle $\boldsymbol{\theta}$ can be expressed as

\[\mathbf{w} = \mathbf{R}(\mathbf{u}, \theta) \mathbf{v},\]

where $\mathbf{R}$ is a $\mathbb{R}^{3\times 3}$ matrix.

This expression can be linearised for it to be included in the linear solver as

\[\delta\mathbf{w} = \frac{\partial}{\partial\theta}\left(\mathbf{R}(\mathbf{u}, \theta_0)\right)\delta\theta\]

The matrix $\mathbf{R}$ is

\[\begin{split}\mathbf{R} = \begin{bmatrix}\cos \theta +u_{x}^{2}\left(1-\cos \theta \right) & u_{x}u_{y}\left(1-\cos \theta \right)-u_{z}\sin \theta & u_{x}u_{z}\left(1-\cos \theta \right)+u_{y}\sin \theta \\ u_{y}u_{x}\left(1-\cos \theta \right)+u_{z}\sin \theta & \cos \theta +u_{y}^{2}\left(1-\cos \theta \right)& u_{y}u_{z}\left(1-\cos \theta \right)-u_{x}\sin \theta \\ u_{z}u_{x}\left(1-\cos \theta \right)-u_{y}\sin \theta & u_{z}u_{y}\left(1-\cos \theta \right)+u_{x}\sin \theta & \cos \theta +u_{z}^{2}\left(1-\cos \theta \right)\end{bmatrix},\end{split}\]

and its linearised expression becomes

\[\begin{split}\frac{\partial}{\partial\theta}\left(\mathbf{R}(\mathbf{u}, \theta_0)\right) = \begin{bmatrix} -\sin \theta +u_{x}^{2}\sin \theta \mathbf{v}_1 + u_{x}u_{y}\sin \theta-u_{z} \cos \theta \mathbf{v}_2 + u_{x}u_{z}\sin \theta +u_{y}\cos \theta \mathbf{v}_3 \\ u_{y}u_{x}\sin \theta+u_{z}\cos \theta\mathbf{v}_1 -\sin \theta +u_{y}^{2}\sin \theta\mathbf{v}_2 + u_{y}u_{z}\sin \theta-u_{x}\cos \theta\mathbf{v}_3 \\ u_{z}u_{x}\sin \theta-u_{y}\cos \theta\mathbf{v}_1 + u_{z}u_{y}\sin \theta+u_{x}\cos \theta\mathbf{v}_2 -\sin \theta +u_{z}^{2}\sin\theta\mathbf{v}_3\end{bmatrix}_{\theta=\theta_0}\end{split}\]

and is of dimension $\mathbb{R}^{3\times 1}$.

param u: Arbitrary rotation axis
type u: numpy.ndarray
param theta: Rotation angle (radians)
type theta: float
param v: Vector to rotate
type v: numpy.ndarray
returns: Linearised rotation vector of dimensions $\mathbb{R}^{3\times 1}$.
rtype: numpy.ndarray

LinearAeroelastic¶

class sharpy.linear.assembler.linearaeroelastic.LinearAeroelastic[source]¶

Assemble a linearised aeroelastic system

The aeroelastic system can be seen as the coupling between a linearised aerodynamic system (System 1) and a linearised beam system (System 2).

The coupled system retains inputs and outputs from both systems such that

\[\mathbf{u} = [\mathbf{u}_1;\, \mathbf{u}_2]\]

and the outputs are also ordered in a similar fashion

\[\mathbf{y} = [\mathbf{y}_1;\, \mathbf{y}_2]\]

Reference the individual systems for the particular ordering of the respective input and output variables.

The settings that this solver accepts are given by a dictionary, with the following key-value pairs:

Name	Type	Description	Default
`aero_settings`	`dict`	Linear UVLM settings	`None`
`beam_settings`	`dict`	Linear Beam settings	`None`
`uvlm_filename`	`str`	Path to .data.h5 file containing UVLM/ROM state space to load
`track_body`	`bool`	UVLM inputs and outputs projected to coincide with lattice at linearisation	`True`
`use_euler`	`bool`	Parametrise orientations in terms of Euler angles	`True`

assemble()[source]¶

Assembly of the linearised aeroelastic system.

The UVLM state-space system has already been assembled. Prior to assembling the beam’s first order state-space, the damping and stiffness matrices have to be modified to include the damping and stiffenning terms that arise from the linearisation of the aeordynamic forces with respect to the A frame of reference. See sharpy.linear.src.lin_aeroela.get_gebm2uvlm_gains() for details on the linearisation.

Then the beam is assembled as per the given settings in normalised time if the aerodynamic system has been scaled. The discrete time systems of the UVLM and the beam must have the same time step.

The UVLM inputs and outputs are then projected onto the structural degrees of freedom (obviously with the exception of external gusts and control surfaces). Hence, the gains $\mathbf{K}_{sa}$ and $\mathbf{K}_{as}$ are added to the output and input of the UVLM system, respectively. These gains perform the following relation:

\[\begin{split}\begin{bmatrix}\zeta \\ \zeta' \\ u_g \\ \delta \end{bmatrix} = \mathbf{K}_{as} \begin{bmatrix} \eta \\ \eta' \\ u_g \\ \delta \end{bmatrix} =\end{split}\]

\[\mathbf{N}_{nodes} = \mathbf{K}_{sa} \mathbf{f}_{vertices}\]

If the beam is expressed in modal form, the UVLM is further projected onto the beam’s modes to have the following input/output structure:

Returns:

get_gebm2uvlm_gains(data)[source]¶

Provides:

the gain matrices required to connect the linearised GEBM and UVLM

inputs/outputs

the stiffening and damping factors to be added to the linearised GEBM equations in order to account for non-zero aerodynamic loads at the linearisation point.

The function produces the gain matrices:

Kdisp: gains from GEBM to UVLM grid displacements

Kvel_disp: influence of GEBM dofs displacements to UVLM grid velocities.

Kvel_vel: influence of GEBM dofs displacements to UVLM grid displacements.

Kforces (UVLM->GEBM) dimensions are the transpose than the
Kdisp and Kvel* matrices. Hence, when allocation this term, ii and jj indices will unintuitively refer to columns and rows,

respectively.

And the stiffening/damping terms accounting for non-zero aerodynamic forces at the linearisation point:

Kss: stiffness factor (flexible dof -> flexible dof) accounting for non-zero forces at the linearisation point.

Csr: damping factor (rigid dof -> flexible dof)

Crs: damping factor (flexible dof -> rigid dof)

Crr: damping factor (rigid dof -> rigid dof)

Stiffening and damping related terms due to the non-zero aerodynamic forces at the linearisation point:

\[\mathbf{F}_{A,n} = C^{AG}(\mathbf{\chi})\sum_j \mathbf{f}_{G,j} \rightarrow \delta\mathbf{F}_{A,n} = C^{AG}_0 \sum_j \delta\mathbf{f}_{G,j} + \frac{\partial}{\partial\chi}(C^{AG}\sum_j \mathbf{f}_{G,j}^0)\delta\chi\]

The term multiplied by the variation in the quaternion, $\delta\chi$, couples the forces with the rigid body equations and becomes part of $\mathbf{C}_{sr}$.

Similarly, the linearisation of the moments results in expression that contribute to the stiffness and damping matrices.

\[\mathbf{M}_{B,n} = \sum_j \tilde{X}_B C^{BA}(\Psi)C^{AG}(\chi)\mathbf{f}_{G,j}\]

\[\delta\mathbf{M}_{B,n} = \sum_j \tilde{X}_B\left(C_0^{BG}\delta\mathbf{f}_{G,j} + \frac{\partial}{\partial\Psi}(C^{BA}\delta\mathbf{f}^0_{A,j})\delta\Psi + \frac{\partial}{\partial\chi}(C^{BA}_0 C^{AG} \mathbf{f}_{G,j})\delta\chi\right)\]

The linearised equations of motion for the geometrically exact beam model take the input term $\delta \mathbf{Q}_n = \{\delta\mathbf{F}_{A,n},\, T_0^T\delta\mathbf{M}_{B,n}\}$, which means that the moments should be provided as $T^T(\Psi)\mathbf{M}_B$ instead of $\mathbf{M}_A = C^{AB}\mathbf{M}_B$, where $T(\Psi)$ is the tangential operator.

\[\delta(T^T\mathbf{M}_B) = T^T_0\delta\mathbf{M}_B + \frac{\partial}{\partial\Psi}(T^T\delta\mathbf{M}_B^0)\delta\Psi\]

is the linearised expression for the moments, where the first term would correspond to the input terms to the beam equations and the second arises due to the non-zero aerodynamic moment at the linearisation point and must be subtracted (since it comes from the forces) to form part of $\mathbf{K}_{ss}$. In addition, the $\delta\mathbf{M}_B$ term depends on both $\delta\Psi$ and $\delta\chi$, therefore those terms would also contribute to $\mathbf{K}_{ss}$ and $\mathbf{C}_{sr}$, respectively.

The contribution from the total forces and moments will be accounted for in $\mathbf{C}_{rr}$ and $\mathbf{C}_{rs}$.

\[\delta\mathbf{F}_{tot,A} = \sum_n\left(C^{GA}_0 \sum_j \delta\mathbf{f}_{G,j} + \frac{\partial}{\partial\chi}(C^{AG}\sum_j \mathbf{f}_{G,j}^0)\delta\chi\right)\]

Therefore, after running this method, the beam matrices will be updated as:

>>> K_beam[:flex_dof, :flex_dof] += Kss
>>> C_beam[:flex_dof, -rigid_dof:] += Csr
>>> C_beam[-rigid_dof:, :flex_dof] += Crs
>>> C_beam[-rigid_dof:, -rigid_dof:] += Crr

Track body option

The track_body setting restricts the UVLM grid to linear translation motions and therefore should be used to ensure that the forces are computed using the reference linearisation frame.

The UVLM and beam are linearised about a reference equilibrium condition. The UVLM is defined in the inertial reference frame while the beam employs the body attached frame and therefore a projection from one frame onto another is required during the coupling process.

However, the inputs to the UVLM (i.e. the lattice grid coordinates) are obtained from the beam deformation which is expressed in A frame and therefore the grid coordinates need to be projected onto the inertial frame G. As the beam rotates, the projection onto the G frame of the lattice grid coordinates will result in a grid that is not coincident with that at the linearisation reference and therefore the grid coordinates must be projected onto the original frame, which will be referred to as U. The transformation between the inertial frame G and the U frame is a function of the rotation of the A frame and the original position:

\[C^{UG}(\chi) = C^{GA}(\chi_0)C^{AG}(\chi)\]

Therefore, the grid coordinates obtained in A frame and projected onto the G frame can be transformed to the U frame using

\[\zeta_U = C^{UG}(\chi) \zeta_G\]

which allows the grid lattice coordinates to be projected onto the original linearisation frame.

In a similar fashion, the output lattice vertex forces of the UVLM are defined in the original linearisation frame U and need to be transformed onto the inertial frame G prior to projecting them onto the A frame to use them as the input forces to the beam system.

\[\boldsymbol{f}_G = C^{GU}(\chi)\boldsymbol{f}_U\]

The linearisation of the above relations lead to the following expressions that have to be added to the coupling matrices:

Kdisp_vel terms:

\[\delta\boldsymbol{\zeta}_U= C^{GA}_0 \frac{\partial}{\partial \boldsymbol{\chi}} \left(C^{AG}\boldsymbol{\zeta}_{G,0}\right)\delta\boldsymbol{\chi} + \delta\boldsymbol{\zeta}_G\]

Kvel_vel terms:

\[\delta\dot{\boldsymbol{\zeta}}_U= C^{GA}_0 \frac{\partial}{\partial \boldsymbol{\chi}} \left(C^{AG}\dot{\boldsymbol{\zeta}}_{G,0}\right)\delta\boldsymbol{\chi} + \delta\dot{\boldsymbol{\zeta}}_G\]

The transformation of the forces and moments introduces terms that are functions of the orientation and are included as stiffening and damping terms in the beam’s matrices:

Csr damping terms relating to translation forces:

\[C_{sr}^{tra} -= \frac{\partial}{\partial\boldsymbol{\chi}} \left(C^{GA} C^{AG}_0 \boldsymbol{f}_{G,0}\right)\delta\boldsymbol{\chi}\]

Csr damping terms related to moments:

\[C_{sr}^{rot} -= T^\top\widetilde{\mathbf{X}}_B C^{BG} \frac{\partial}{\partial\boldsymbol{\chi}} \left(C^{GA} C^{AG}_0 \boldsymbol{f}_{G,0}\right)\delta\boldsymbol{\chi}\]

The track_body setting.

When track_body is enabled, the UVLM grid is no longer coincident with the inertial reference frame throughout the simulation but rather it is able to rotate as the A frame rotates. This is to simulate a free flying vehicle, where, for instance, the orientation does not affect the aerodynamics. The UVLM defined in this frame of reference, named U, satisfies the following convention:

The U frame is coincident with the G frame at the time of linearisation.

The U frame rotates as the A frame rotates.

Transformations related to the U frame of reference:

The angle between the U frame and the A frame is always constant and equal to $\boldsymbol{\Theta}_0$.

The angle between the A frame and the G frame is $\boldsymbol{\Theta}=\boldsymbol{\Theta}_0 + \delta\boldsymbol{\Theta}$

The projection of a vector expressed in the G frame onto the U frame is expressed by:

\[\boldsymbol{v}^U = C^{GA}_0 C^{AG} \boldsymbol{v}^G\]

The reverse, a projection of a vector expressed in the U frame onto the G frame, is expressed by

\[\boldsymbol{v}^U = C^{GA} C^{AG}_0 \boldsymbol{v}^U\]

The effect this has on the aeroelastic coupling between the UVLM and the structural dynamics is that the orientation and change of orientation of the vehicle has no effect on the aerodynamics. The aerodynamics are solely affected by the contribution of the 6-rigid body velocities (as well as the flexible DOFs velocities).

update(u_infty)[source]¶

Updates the aeroelastic scaled system with the new reference velocity.

Only the beam equations need updating since the only dependency in the forward flight velocity resides there.

Parameters: u_infty (float) – New reference velocity
Returns: Updated aeroelastic state-space system
Return type: sharpy.linear.src.libss.ss

Linear State Beam Element Class¶

Linear State Beam Element Class

LinearBeam¶

class sharpy.linear.assembler.linearbeam.LinearBeam[source]¶

State space member

Define class for linear state-space realisation of GEBM flexible-body equations from SHARPy timestep_info class and with the nonlinear structural information.

State-space models can be defined in continuous or discrete time (dt required). Modal projection, either on the damped or undamped modal shapes, is also avaiable.

Notes on the settings:

modal_projection={True,False}: determines whether to project the states
onto modal coordinates. Projection over damped or undamped modal shapes can be obtained selecting:

proj_modes = {'damped','undamped'}

while

inout_coords={'modes','nodal'}

determines whether the modal state-space inputs/outputs are modal coords or nodal degrees-of-freedom. If modes is selected, the Kin and Kout gain matrices are generated to transform nodal to modal dofs
dlti={True,False}: if true, generates discrete-time system.
The continuous to discrete transformation method is determined by:
discr_method={ 'newmark',  # Newmark-beta
                    'zoh',              # Zero-order hold
                    'bilinear'} # Bilinear (Tustin) transformation
DLTIs can be obtained directly using the Newmark-$\beta$ method

discr_method='newmark' newmark_damp=xx with xx<<1.0

for full-states descriptions (modal_projection=False) and modal projection over the undamped structural modes (modal_projection=True and proj_modes). The Zero-order holder and bilinear methods, instead, work in all descriptions, but require the continuous state-space equations.

The settings that this solver accepts are given by a dictionary, with the following key-value pairs:

Name	Type	Description	Default	Options
`modal_projection`	`bool`	Use modal projection	`True`
`inout_coords`	`str`	Beam state space input/output coordinates	`nodes`	`nodes`, `modes`
`num_modes`	`int`	Number of modes to retain	`10`
`discrete_time`	`bool`	Assemble beam in discrete time	`True`
`dt`	`float`	Discrete time system integration time step	`0.001`
`proj_modes`	`str`	Use `undamped` or `damped` modes	`undamped`	`damped`, `undamped`
`discr_method`	`str`	Discrete time assembly system method:	`newmark`	`newmark`, `zoh`, `bilinear`
`newmark_damp`	`float`	Newmark damping value. For systems assembled using `newmark`	`0.0001`
`use_euler`	`bool`	Use euler angles for rigid body parametrisation	`False`
`print_info`	`bool`	Display information on screen	`True`
`gravity`	`bool`	Linearise gravitational forces	`False`
`remove_dofs`	`list(str)`	Remove desired degrees of freedom	`[]`	`eta`, `V`, `W`, `orient`
`remove_sym_modes`	`bool`	Remove symmetric modes if wing is clamped	`False`

assemble(t_ref=None)[source]¶

Assemble the beam state-space system.

Parameters: t_ref (float) – Scaling factor to non-dimensionalise the beam’s time step.

Returns:

remove_symmetric_modes()[source]¶

Removes symmetric modes when the wing is clamped at the midpoint.

It will force the wing tip displacements in z to be postive for all modes.

Updates the mode shapes matrix, the natural frequencies and the number of modes.

unpack_ss_vector(x_n, u_n, struct_tstep)[source]¶

Warning

Under development. Missing:

Accelerations
Double check the cartesian rotation vector
Tangential operator for the moments

Takes the state $x = [\eta, \dot{\eta}]$ and input vectors $u = N$ of a linearised beam and returns a SHARPy timestep instance, including the reference values.

Parameters

x_n (np.ndarray) – Structural beam state vector in nodal space
y_n (np.ndarray) – Beam input vector (nodal forces)
struct_tstep (utils.datastructures.StructTimeStepInfo) – Reference timestep used for linearisation

Returns

new timestep with linearised values added to the reference value

Return type

utils.datastructures.StructTimeStepInfo

LinearGustGenerator¶

class sharpy.linear.assembler.lineargustassembler.LinearGustGenerator[source]¶: Reduces the entire gust field input to a user-defined set of more comprehensive inputs

Linear UVLM State Space System¶

Linear UVLM State Space System

LinearUVLM¶

class sharpy.linear.assembler.linearuvlm.LinearUVLM[source]¶

Linear UVLM System Assembler

Produces state-space model of the form

\[\begin{split}\mathbf{x}_{n+1} &= \mathbf{A}\,\mathbf{x}_n + \mathbf{B} \mathbf{u}_{n+1} \\ \mathbf{y}_n &= \mathbf{C}\,\mathbf{x}_n + \mathbf{D} \mathbf{u}_n\end{split}\]

where the state, inputs and outputs are:

\[\mathbf{x}_n = \{ \delta \mathbf{\Gamma}_n,\, \delta \mathbf{\Gamma_{w_n}},\, \Delta t\,\delta\mathbf{\Gamma}'_n,\, \delta\mathbf{\Gamma}_{n-1} \}\]

\[\mathbf{u}_n = \{ \delta\mathbf{\zeta}_n,\, \delta\mathbf{\zeta}'_n,\, \delta\mathbf{u}_{ext,n} \}\]

\[\mathbf{y} = \{\delta\mathbf{f}\}\]

with $\mathbf{\Gamma}\in\mathbb{R}^{MN}$ being the vector of vortex circulations, $\mathbf{\zeta}\in\mathbb{R}^{3(M+1)(N+1)}$ the vector of vortex lattice coordinates and $\mathbf{f}\in\mathbb{R}^{3(M+1)(N+1)}$ the vector of aerodynamic forces and moments. Note that $(\bullet)'$ denotes a derivative with respect to time.

Note that the input is atypically defined at time n+1. If the setting remove_predictor = True the predictor term u_{n+1} is eliminated through the change of state[1]:

\[\begin{split}\mathbf{h}_n &= \mathbf{x}_n - \mathbf{B}\,\mathbf{u}_n \\\end{split}\]

such that:

\[\begin{split}\mathbf{h}_{n+1} &= \mathbf{A}\,\mathbf{h}_n + \mathbf{A\,B}\,\mathbf{u}_n \\ \mathbf{y}_n &= \mathbf{C\,h}_n + (\mathbf{C\,B}+\mathbf{D})\,\mathbf{u}_n\end{split}\]

which only modifies the equivalent $\mathbf{B}$ and $\mathbf{D}$ matrices.

The integr_order setting refers to the finite differencing scheme used to calculate the bound circulation derivative with respect to time $\dot{\mathbf{\Gamma}}$. A first order scheme is used when integr_order == 1

\[\dot{\mathbf{\Gamma}}^{n+1} = \frac{\mathbf{\Gamma}^{n+1}-\mathbf{\Gamma}^n}{\Delta t}\]

If integr_order == 2 a higher order scheme is used (but it isn’t exactly second order accurate [1]).

\[\dot{\mathbf{\Gamma}}^{n+1} = \frac{3\mathbf{\Gamma}^{n+1}-4\mathbf{\Gamma}^n + \mathbf{\Gamma}^{n-1}} {2\Delta t}\]

References

[1] Franklin, GF and Powell, JD. Digital Control of Dynamic Systems, Addison-Wesley Publishing Company, 1980

[2] Maraniello, S., & Palacios, R.. State-Space Realizations and Internal Balancing in Potential-Flow Aerodynamics with Arbitrary Kinematics. AIAA Journal, 57(6), 1–14. 2019. https://doi.org/10.2514/1.J058153

The settings that this solver accepts are given by a dictionary, with the following key-value pairs:

Name	Type	Description	Default	Options
`dt`	`float`	Time step	`0.1`
`integr_order`	`int`	Integration order of the circulation derivative.	`2`	`1`, `2`
`ScalingDict`	`dict`	Dictionary of scaling factors to achieve normalised UVLM realisation.	`{}`
`remove_predictor`	`bool`	Remove the predictor term from the UVLM equations	`True`
`use_sparse`	`bool`	Assemble UVLM plant matrix in sparse format	`True`
`density`	`float`	Air density	`1.225`
`remove_inputs`	`list(str)`	List of inputs to remove. `u_gust` to remove external velocity input.	`[]`	`u_gust`
`gust_assembler`	`str`	Selected linear gust assembler.		`leading_edge`
`rom_method`	`list(str)`	List of model reduction methods to reduce UVLM.	`[]`
`rom_method_settings`	`dict`	Dictionary with settings for the desired ROM methods, where the name of the ROM method is the key to the dictionary	`{}`
`vortex_radius`	`float`	Distance below which inductions are not computed	`sharpy.utils.constants.vortex_radius_def`
`cfl1`	`bool`	If it is `True`, it assumes that the discretisation complies with CFL=1	`True`

The settings that this solver accepts are given by a dictionary, with the following key-value pairs:

Name	Type	Description	Default
`length`	`float`	Reference length to be used for UVLM scaling	`1.0`
`speed`	`float`	Reference speed to be used for UVLM scaling	`1.0`
`density`	`float`	Reference density to be used for UVLM scaling	`1.0`

assemble(track_body=False, wake_prop_settings=None)[source]¶

Assembles the linearised UVLM system, removes the desired inputs and adds linearised control surfaces (if present).

With all possible inputs present, these are ordered as

\[\mathbf{u} = [\boldsymbol{\zeta},\,\dot{\boldsymbol{\zeta}},\,\mathbf{w},\,\delta]\]

Control surface inputs are ordered last as:

\[[\delta_1, \delta_2, \dots, \dot{\delta}_1, \dot{\delta_2}]\]

remove_inputs(remove_list=<class 'list'>)[source]¶

Remove certain inputs from the input vector

To do:

Support for block UVLM

Parameters: remove_list (list) – Inputs to remove

unpack_input_vector(u_n)[source]¶

Unpacks the input vector into the corresponding grid coordinates, velocities and external velocities.

Parameters: u_n (np.ndarray) – UVLM input vector. May contain control surface deflections and external velocities.
Returns: Tuple containing zeta, zeta_dot and u_ext, accounting for the effect of control surfaces.
Return type: tuple

unpack_ss_vector(data, x_n, aero_tstep, track_body=False)[source]¶

Transform column vectors used in the state space formulation into SHARPy format

The column vectors are transformed into lists with one entry per aerodynamic surface. Each entry contains a matrix with the quantities at each grid vertex.

\[\mathbf{y}_n \longrightarrow \mathbf{f}_{aero}\]

\[\mathbf{x}_n \longrightarrow \mathbf{\Gamma}_n,\, \mathbf{\Gamma_w}_n,\, \mathbf{\dot{\Gamma}}_n\]

If the track_body option is on, the output forces are projected from the linearization frame, to the G frame. Note that the linearisation frame is:

equal to the FoR G at time 0 (linearisation point)

rotates as the body frame specified in the track_body_number

Parameters

y_n (np.ndarray) – Column output vector of linear UVLM system
x_n (np.ndarray) – Column state vector of linear UVLM system
u_n (np.ndarray) – Column input vector of linear UVLM system
aero_tstep (AeroTimeStepInfo) – aerodynamic timestep information class instance

Returns

Tuple containing:

forces (list):
Aerodynamic forces in a list with n_surf entries. Each entry is a (6, M+1, N+1) matrix, where the first 3 indices correspond to the components in x, y and z. The latter 3 are zero.

gamma (list):
Bound circulation list with n_surf entries. Circulation is stored in an (M+1, N+1) matrix, corresponding to the panel vertices.

gamma_dot (list):
Bound circulation derivative list with n_surf entries. Circulation derivative is stored in an (M+1, N+1) matrix, corresponding to the panel vertices.

gamma_star (list):
Wake (free) circulation list with n_surf entries. Wake circulation is stored in an (M_star+1, N+1) matrix, corresponding to the panel vertices of the wake.

Return type

tuple

Linearised System Source Code¶

Assembly of linearised UVLM system¶

Maraniello, 25 May 2018

Includes:

Boundary conditions methods:
- AICs: allocate aero influence coefficient matrices of multi-surfaces configurations
- nc_dqcdzeta_Sin_to_Sout: derivative matrix of nc*dQ/dzeta where Q is the induced velocity at the bound collocation points of one surface to another.
- nc_dqcdzeta_coll: assembles nc_dqcdzeta_coll_Sin_to_Sout matrices in multi-surfaces configurations
- uc_dncdzeta: assemble derivative matrix dnc/dzeta*Uc at bound collocation points

AICs¶

Given a list of bound (Surfs) and wake (Surfs_star) instances of surface.AeroGridSurface, returns the list of AIC matrices in the format:

AIC_list[ii][jj] contains the AIC from the bound surface Surfs[jj] to

Surfs[ii]. - AIC_star_list[ii][jj] contains the AIC from the wake surface Surfs[jj] to Surfs[ii].

dfqsdgamma_vrel0¶

Assemble derivative of quasi-steady force w.r.t. gamma with fixed relative velocity - the changes in induced velocities due to gamma are not accounted for. The routine exploits the get_joukovski_qs method insude the AeroGridSurface class

dfqsduinput¶

Assemble derivative of quasi-steady force w.r.t. external input velocity.

dfqsdvind_gamma¶

Assemble derivative of quasi-steady force w.r.t. induced velocities changes due to gamma. Note: the routine is memory consuming but avoids unnecessary computations.

dfqsdvind_zeta¶

Assemble derivative of quasi-steady force w.r.t. induced velocities changes due to zeta.

dfqsdzeta_omega¶

Assemble derivative of quasi-steady force w.r.t. to zeta The contribution implemented is related with the omega x zeta term call: Der_list = dfqsdzeta_omega(Surfs,Surfs_star)

dfqsdzeta_vrel0¶

Assemble derivative of quasi-steady force w.r.t. zeta with fixed relative velocity - the changes in induced velocities due to zeta over the surface inducing the velocity are not accounted for. The routine exploits the available relative velocities at the mid-segment points

dfunstdgamma_dot¶

Computes derivative of unsteady aerodynamic force with respect to changes in circulation.

Note: the function also checks that the first derivative of the circulation at the linearisation point is null. If not, a further contribution to the added mass, depending on the changes in panel area and normal, arises and needs to be implemented.

dvinddzeta¶

Produces derivatives of induced velocity by Surf_in w.r.t. the zetac point. Derivatives are divided into those associated to the movement of zetac, and to the movement of the Surf_in vertices (DerVert).

If Surf_in is bound (IsBound==True), the circulation over the TE due to the wake is not included in the input.

If Surf_in is a wake (IsBound==False), derivatives w.r.t. collocation points are computed ad the TE contribution on DerVert. In this case, the chordwise paneling Min_bound of the associated input is required so as to calculate Kzeta and correctly allocate the derivative matrix.

The output derivatives are: - Dercoll: 3 x 3 matrix - Dervert: 3 x 3*Kzeta (if Surf_in is a wake, Kzeta is that of the bound)

Warning: zetac must be contiguously stored!

dvinddzeta_cpp¶

eval_panel_cpp¶

nc_domegazetadzeta¶

Produces a list of derivative matrix d(omaga x zeta)/dzeta, where omega is the rotation speed of the A FoR, ASSUMING constant panel norm.

Each list is such that: - the ii-th element is associated to the ii-th bound surface collocation point, and will contain a sub-list such that:

the j-th element of the sub-list is the dAIC_dzeta matrices w.r.t. the

zeta d.o.f. of the j-th bound surface.

Hence, DAIC*[ii][jj] will have size K_ii x Kzeta_jj

call: ncDOmegaZetavert = nc_domegazetadzeta(Surfs,Surfs_star)

nc_dqcdzeta¶

Produces a list of derivative matrix

\[\frac{\partial(\mathcal{A}\boldsymbol{\Gamma}_0)}{\partial\boldsymbol{\zeta}}\]

where $\mathcal{A}$ is the aerodynamic influence coefficient matrix at the bound surfaces collocation point, assuming constant panel norm.

Each list is such that:

the ii-th element is associated to the ii-th bound surface collocation point, and will contain a sub-list such that:

the j-th element of the sub-list is the dAIC_dzeta matrices w.r.t. the zeta d.o.f. of the j-th bound surface.

Hence, DAIC*[ii][jj] will have size K_ii x Kzeta_jj

If Merge is True, the derivatives due to collocation points movement are added to Dvert to minimise storage space.

To do:

Dcoll is highly sparse, exploit?

nc_dqcdzeta_Sin_to_Sout¶

Computes derivative matrix of: nc*dQ/dzeta

where Q is the induced velocity induced by bound surface Surf_in onto bound surface Surf_out. The panel normals of Surf_out are constant.

The input/output are: - Der_coll of size (Kout,3*Kzeta_out): derivative due to the movement of collocation point on Surf_out. - Der_vert of size:

(Kout,3*Kzeta_in) if Surf_in_bound is True

(Kout,3*Kzeta_bound_in) if Surf_in_bound is False; Kzeta_bound_in is

the number of vertices in the bound surface of whom Surf_out is the wake.

Note that: - if Surf_in_bound is False, only the TE movement contributes to Der_vert. - if Surf_in_bound is False, the allocation of Der_coll could be speed-up by scanning only the wake segments along the chordwise direction, as on the others the net circulation is null.

test_wake_prop_term¶

Test allocation of single term of wake propagation matrix

uc_dncdzeta¶

Build derivative of

\[\boldsymbol{u}_c\frac{\partial\boldsymbol{n}_c}{\partial\boldsymbol{zeta}}\]

where $\boldsymbol{u}_c$ is the total velocity at the collocation points.

param Surf: the input can also be a list of surface.AerogridSurface
type Surf: surface.AerogridSurface

References

:module:`linear.develop_sym.linsum_Wnc`
:module:`lib_ucdncdzeta`

wake_prop¶

Assembly of wake propagation matrices, in sparse or dense matrices format

Note

Wake propagation matrices are very sparse. Nonetheless, allocation in dense format (from numpy.zeros) or sparse does not have important differences in terms of cpu time and memory used as numpy.zeros does not allocate memory until this is accessed

param MS: MultiSurface instance
type MS: MultiSurface
param use_sparse: Use sparse matrices
type use_sparse: bool (optional)
param sparse_format: Use either csc or lil format
type sparse_format: str (optional)
param settings: Dictionary with aerodynamic settings containing: cfl1 (bool): Defines if the wake shape complies with CFL=1 dt (float): time step
type settings: dict (optional)

wake_prop_from_dimensions¶

Same as ``wake_prop’’ but using the dimensions directly

Mapping methods for bound surface panels¶

Maraniello, 19 May 2018

AeroGridMap¶

class sharpy.linear.src.gridmapping.AeroGridMap(M: number of chord-wise, N: number of span-wise)[source]¶

Produces mapping between panels, segment and vertices of a surface. Grid elements are identified through the indices (m,n), where:

m: chordwise index

n: spanwise index

The same indexing is applied to panel, vertices and segments.

Elements: - panels=(M,N) - vertices=(M+1,N+1) - segments: these are divided in segments developing along the chordwise and spanwise directions.

chordwise: (M,N+1)

spanwise: (M+1,N)

Mapping structures: - Mpv: for each panel (mp,np) returns the chord/span-wise indices of its vertices, (mv,nv). This has size (M,N,4,2) - Mps: maps each panel (mp,np) to the ii-th segment. This has size (M,N,4,2)

# - Mps_extr: for each panel (m,n) returns the indices of the extrema of each side # of the panel.

Note: - mapping matrices are stored as np.int16 or np.int32 arrays

from_panel_to_segments(m: chordwise index, n: spanwise index)[source]¶: For each panel (m,n) it provides the ms,ns indices of each segment.

from_panel_to_vertices(m: chordwise index, n: spanwise index)[source]¶: From panel of indices (m,n) to indices of vertices

from_vertex_to_panel(m: chordwise index, n: spanwise index)[source]¶

Returns the panel for which the vertex is locally numbered as 0,1,2,3. Returns a (4,2) array such that its elements are:

[vv_local,(m,n) of panel]

where vv_local is the local verteix number.

Important: indices -1 are possible is the vertex does not have local index 0,1,2 or 3 with respect to any panel.

map_panels_to_segments()[source]¶

Mapping from panel of segments. self.Mpv is a (M,N,4,2) array such that:

[m, n, local_segment_number,
chordwise/spanwise index of segment,]

map_panels_to_vertices()[source]¶: Mapping from panel of vertices. self.Mpv is a (M,N,4,2) array such that its element are:

[m, n, local_vertex_number, spanwise/chordwise indices of vertex]

map_panels_to_vertices_1D_scalar()[source]¶

Mapping: - FROM: the index of a scalar quantity defined at panel collocation point and stored in 1D array. - TO: index of a scalar quantity defined at vertices and stored in 1D

The Mpv1d_scalar has size (K,4) where:: [1d index of panel, index of vertex 0,1,2 or 3]

map_vertices_to_panels()[source]¶

Maps from vertices to panels. Produces a (M+1,N+1,4,2) array, associating vertices to panels. Its elements are:

[m vertex,

n vertex,

vertex local index,
chordwise/spanwise panel indices]

map_vertices_to_panels_1D_scalar()[source]¶

Mapping: - FROM: the index of a scalar quantity defined at vertices and stored in 1D array. - TO: index of a scalar quantity defined at panels and stored in 1D

The Mpv1d_scalar has size (Kzeta,4) where:: [1d index of vertex, index of vertex 0,1,2 or 3 w.r.t. panel]

Defines interpolation methods (geometrically-exact) and matrices (linearisation)¶

Defines interpolation methods (geometrically-exact) and matrices (linearisation) S. Maraniello, 20 May 2018

get_Wnv_vector¶

Provide projection matrix from nodal velocities to normal velocity at collocation points

get_Wvc_scalar¶

Produce scalar interpolation matrix Wvc for state-space realisation.

Important: this will not work for coordinates extrapolation, as it would require information about the panel size. It works for other forces/scalar quantities extrapolation. It assumes the quantity at the collocation point is determined proportionally to the weight associated to each vertex and obtained through get_panel_wcv.

get_panel_wcv¶

Produces a compact array with weights for bilinear interpolation, where aN,aM in [0,1] are distances in the chordwise and spanwise directions such that:

(aM,aN)=(0,0) –> quantity at vertex 0

(aM,aN)=(1,0) –> quantity at vertex 1

(aM,aN)=(1,1) –> quantity at vertex 2

(aM,aN)=(0,1) –> quantity at vertex 3

Induced Velocity Derivatives¶

Calculate derivatives of induced velocity.

Methods:

eval_seg_exp and eval_seg_exp_loop: profide ders in format
[Q_{x,y,z},ZetaPoint_{x,y,z}]

and use fully-expanded analytical formula.
eval_panel_exp: iterates through whole panel
eval_seg_comp and eval_seg_comp_loop: profide ders in format
[Q_{x,y,z},ZetaPoint_{x,y,z}]

and use compact analytical formula.

Dvcross_by_skew3d¶

eval_panel_comp¶

eval_panel_cpp¶

eval_panel_exp¶

eval_panel_fast¶

eval_panel_fast_coll¶

eval_seg_comp_loop¶

eval_seg_exp¶

eval_seg_exp_loop¶

Induced Velocity Derivatives with respect to Panel Normal¶

Calculate derivative of

\[\boldsymbol{u}_c\frac{\partial\boldsymbol{n}_c}{\partial\boldsymbol{zeta}}\]

with respect to local panel coordinates.

eval¶

Returns a 4 x 3 array, containing the derivative of Wnc*Uc w.r.t the panel vertices coordinates.

Fitting Tools Library¶

@author: Salvatore Maraniello

@date: 15 Jan 2018

fitfrd¶

Wrapper for fitfrd (mag=0) and fitfrdmag (mag=1) functions in continuous and discrete time (if ds in input). Input:

kv,yv: frequency array and frequency response N: order for rational function approximation mag=1,0: Flag for determining method to use dt (optional): sampling time for DLTI systems

get_rfa_res¶

Returns magnitude of the residual Yfit-Yv of a RFA approximation at each point kv. The coefficients of the approximations are: - cnum=xv[:Nnum] - cdem=xv[Nnum:] where cnum and cden are as per the ‘rfa’ function.

get_rfa_res_norm¶

Define residual scalar norm of Pade approximation of coefficients cnum=xv[:Nnum] and cden[Nnum:] (see get_rfa_res and rfa function) and time-step ds (if discrete time).

poly_fit¶

Find best II order fitting polynomial from frequency response Yv over the frequency range kv for both continuous (ds=None) and discrete (ds>0) LTI systems.

Input: - kv: frequency points - Yv: frequency response - dyv,ddyv: frequency responses of I and II order derivatives - method=’leastsq’,’dev’: algorithm for minimisation - Bup (only ‘dev’ method): bounds for bv coefficients as per scipy.optimize.differential_evolution. This is a length 3 array.

Important: - this function attributes equal weight to each data-point!

rfa¶

Evaluates over the frequency range kv.the rational function approximation: [cnum[-1] + cnum[-2] z + … + cnum[0] z**Nnum ]/…

[cden[-1] + cden[-2] z + … + cden[0] z**Nden]

where the numerator and denominator polynomial orders, Nnum and Nden, are the length of the cnum and cden arrays and:

z=exp(1.j*kv*ds), with ds sampling time if ds is given (discrete-time

system) - z=1.*kv, if ds is None (continuous time system)

rfa_fit_dev¶

Find best fitting RFA approximation from frequency response Yv over the frequency range kv for both continuous (ds=None) and discrete (ds>0) LTI systems.

The RFA approximation is found through a 2-stage strategy:: a. an evolutionary algoryhtm is run to determine the optimal fitting coefficients b. the search is refined through a least squares algorithm.
and is stopped as soon as:: 1. the maximum absolute error in frequency response of the RFA falls below TolAbs 2. the maximum number of iterations is reached.

Input: - kv: frequency range for approximation - Yv: frequency response vector over kv - TolAbs: maximum admissible absolute difference in frequency response between RFA and original system. - Nnum,Ndem: number of coefficients for Pade approximation. - ds: sampling time for DLTI systems - NtrialMax: maximum number of repetition of global and least square optimisations - Cfbouds: maximum absolute values of coeffieicnts (only for evolutionary algorithm) - OutFull: if False, only outputs optimal coefficients of RFA. Otherwise,

outputs cost and RFA coefficients of each trial.

Output: - cnopt: optimal coefficients (numerator) - cdopt: optimal coefficients (denominator)

Important: - this function has the same objective as fitfrd in matwrapper module. While generally slower, the global optimisation approach allows to verify the results from fitfrd.

rfa_mimo¶

Given the frequency response of a MIMO DLTI system, this function returns the A,B,C,D matrices associated to the rational function approximation of the original system.

Input: - Yfull: frequency response (as per libss.freqresp) of full size system over the frequencies kv. - kv: array of frequencies over which the RFA approximation is evaluated. - tolAbs: absolute tolerance for the rfa fitting - Nnum: number of numerator coefficients for RFA - Nden: number of denominator coefficients for RFA - NtrialMax: maximum number of attempts - method=[‘intependent’]. Method used to produce the system:

intependent: each input-output combination is treated separately. The

resulting system is a collection of independent SISO DLTIs

rfader¶

Evaluates over the frequency range kv.the derivative of order m of the rational function approximation: [cnum[-1] + cnum[-2] z + … + cnum[0] z**Nnum ]/…

[cden[-1] + cden[-2] z + … + cden[0] z**Nden]

where the numerator and denominator polynomial orders, Nnum and Nden, are the length of the cnum and cden arrays and:

z=exp(1.j*kv*ds), with ds sampling time if ds is given (discrete-time

system) - z=1.*kv, if ds is None (continuous time system)

Collect tools to manipulate sparse and/or mixed dense/sparse matrices.¶

Collect tools to manipulate sparse and/or mixed dense/sparse matrices.

author: S. Maraniello date: Dec 2018

Comment: manipulating large linear system may require using both dense and sparse matrices. While numpy/scipy automatically handle most operations between mixed dense/sparse arrays, some (e.g. dot product) require more attention. This library collects methods to handle these situations.

Classes: scipy.sparse matrices are wrapped so as to ensure compatibility with numpy arrays upon conversion to dense. - csc_matrix: this is a wrapper of scipy.csc_matrix. - SupportedTypes: types supported for operations - WarningTypes: due to some bugs in scipy (v.1.1.0), sum (+) operations between np.ndarray and scipy.sparse matrices can result in numpy.matrixlib.defmatrix.matrix types. This list contains such undesired types that can result from dense/sparse operations and raises a warning if required. (b) convert these types into numpy.ndarrays.

Methods: - dot: handles matrix dot products across different types. - solve: solves linear systems Ax=b with A and b dense, sparse or mixed. - dense: convert matrix to numpy array

Warning: - only sparse types into SupportedTypes are supported!

To Do: - move these methods into an algebra module?

block_dot¶

dot product between block matrices.

Inputs: A, B: are nested lists of dense/sparse matrices of compatible shape for block matrices product. Empty blocks can be defined with None. (see numpy.block)

block_matrix_dot_vector¶

dot product between block matrix and block vector

Inputs: A, v: are nested lists of dense/sparse matrices of compatible shape for block matrices product. Empty blocks can be defined with None. (see numpy.block)

block_sum¶

dot product between block matrices.

Inputs: A, B: are nested lists of dense/sparse matrices of compatible shape for block matrices product. Empty blocks can be defined with None. (see numpy.block)

csc_matrix¶

class sharpy.linear.src.libsparse.csc_matrix(*args: Any, **kwargs: Any)[source]¶

Wrapper of scipy.csc_matrix that ensures best compatibility with numpy.ndarray. The following methods have been overwritten to ensure that numpy.ndarray are returned instead of numpy.matrixlib.defmatrix.matrix.

todense

_add_dense

Warning: this format is memory inefficient to allocate new sparse matrices. Consider using: - scipy.sparse.lil_matrix, which supports slicing, or - scipy.sparse.coo_matrix, though slicing is not supported :(

todense()[source]¶: As per scipy.spmatrix.todense but returns a numpy.ndarray.

dense¶

If required, converts sparse array to dense.

dot¶

Method to compute: C = A*B ,

where * is the matrix product, with dense/sparse/mixed matrices.

The format (sparse or dense) of C is specified through ‘type_out’. If type_out==None, the output format is sparse if both A and B are sparse, dense otherwise.

The following formats are supported: - numpy.ndarray - scipy.csc_matrix

eye_as¶

Produces an identity matrix as per M, in shape and type

solve¶

Wrapper of: numpy.linalg.solve and scipy.sparse.linalg.spsolve

for solution of the linear system A x = b. - if A is a dense numpy array np.linalg.solve is called for solution. Note that if B is sparse, this requires convertion to dense. In this case, solution through LU factorisation of A should be considered to exploit the sparsity of B. - if A is sparse, scipy.sparse.linalg.spsolve is used.

zeros_as¶

Produces an identity matrix as per M, in shape and type

Linear Time Invariant systems¶

Linear Time Invariant systems author: S. Maraniello date: 15 Sep 2017 (still basement…)

Library of methods to build/manipulate state-space models. The module supports the sparse arrays types defined in libsparse.

The module includes:

Classes: - ss: provides a class to build DLTI/LTI systems with full and/or sparse

matrices and wraps many of the methods in these library. Methods include: - freqresp: wraps the freqresp function - addGain: adds gains in input/output. This is not a wrapper of addGain, as the system matrices are overwritten

Methods for state-space manipulation: - couple: feedback coupling. Does not support sparsity - freqresp: calculate frequency response. Supports sparsity. - series: series connection between systems - parallel: parallel connection between systems - SSconv: convert state-space model with predictions and delays - addGain: add gains to state-space model. - join2: merge two state-space models into one. - join: merge a list of state-space models into one. - sum state-space models and/or gains - scale_SS: scale state-space model - simulate: simulates discrete time solution - Hnorm_from_freq_resp: compute H norm of a frequency response - adjust_phase: remove discontinuities from a frequency response

Special Models: - SSderivative: produces DLTI of a numerical derivative scheme - SSintegr: produces DLTI of an integration scheme - build_SS_poly: build state-space model with polynomial terms.

Filtering: - butter

Utilities: - get_freq_from_eigs: clculate frequency corresponding to eigenvalues

Comments: - the module supports sparse matrices hence relies on libsparse.

to do:

remove unnecessary coupling routines
couple function can handle sparse matrices but only outputs dense matrices
- verify if typical coupled systems are sparse
- update routine
- add method to automatically determine whether to use sparse or dense?

Hnorm_from_freq_resp¶

Given a frequency response over a domain kv, this funcion computes the H norms through numerical integration.

Note that if kv[-1]<np.pi/dt, the method assumed gv=0 for each frequency kv[-1]<k<np.pi/dt.

Warning: only use for SISO systems! For MIMO definitions are different

SSconv¶

Convert a DLTI system with prediction and delay of the form:

\[\begin{split}\mathbf{x}_{n+1} &= \mathbf{A\,x}_n + \mathbf{B_0\,u}_n + \mathbf{B_1\,u}_{n+1} + \mathbf{B_{m1}\,u}_{n-1} \\ \mathbf{y}_n &= \mathbf{C\,x}_n + \mathbf{D\,u}_n\end{split}\]

into the state-space form:

\[\begin{split}\mathbf{h}_{n+1} &= \mathbf{A_h\,h}_n + \mathbf{B_h\,u}_n \\ \mathbf{y}_n &= \mathbf{C_h\,h}_n + \mathbf{D_h\,u}_n\end{split}\]

If $\mathbf{B_{m1}}$ is None, the original state is retrieved through

\[\mathbf{x}_n = \mathbf{h}_n + \mathbf{B_1\,u}_n\]

and only the $\mathbf{B}$ and $\mathbf{D}$ matrices are modified.

If $\mathbf{B_{m1}}$ is not None, the SS is augmented with the new state

\[\mathbf{g}_{n} = \mathbf{u}_{n-1}\]

or, equivalently, with the equation

\[\mathbf{g}_{n+1} = \mathbf{u}_n\]

leading to the new form

\[\begin{split}\mathbf{H}_{n+1} &= \mathbf{A_A\,H}_{n} + \mathbf{B_B\,u}_n \\ \mathbf{y}_n &= \mathbf{C_C\,H}_{n} + \mathbf{D_D\,u}_n\end{split}\]

where $\mathbf{H} = (\mathbf{x},\,\mathbf{g})$.

param A: dynamics matrix
type A: np.ndarray
param B0: input matrix for input at current time step n. Set to None if this is zero.
type B0: np.ndarray
param B1: input matrix for input at time step n+1 (predictor term)
type B1: np.ndarray
param C: output matrix
type C: np.ndarray
param D: direct matrix
type D: np.ndarray
param Bm1: input matrix for input at time step n-1 (delay term)
type Bm1: np.ndarray
returns: tuple packed with the state-space matrices $\mathbf{A},\,\mathbf{B},\,\mathbf{C}$ and $\mathbf{D}$.
rtype: tuple

References

Franklin, GF and Powell, JD. Digital Control of Dynamic Systems, Addison-Wesley Publishing Company, 1980

Warning

functions untested for delays (Bm1 != 0)

SSderivative¶

Given a time-step ds, and an single input time history u, this SS model returns the output y=[u,du/ds], where du/dt is computed with second order accuracy.

SSintegr¶

Builds a state-space model of an integrator.

method: Numerical scheme. Available options are:
- 1tay: 1st order Taylor (fwd)
  I[ii+1,:]=I[ii,:] + ds*F[ii,:]
- trap: I[ii+1,:]=I[ii,:] + 0.5*dx*(F[ii,:]+F[ii+1,:])
Note: other option can be constructured if information on derivative of F is available (for e.g.)

addGain¶

Convert input u or output y of a SS DLTI system through gain matrix K. We have the following transformations: - where=’in’: the input dof of the state-space are changed

u_new -> Kmat*u -> SS -> y => u_new -> SSnew -> y

where=’out’: the output dof of the state-space are changed
u -> SS -> y -> Kmat*u -> ynew => u -> SSnew -> ynew
where=’parallel’: the input dofs are changed, but not the output
{u_1 -> SS -> y_1
{ u_2 -> y_2= Kmat*u_2 => u_new=(u_1,u_2) -> SSnew -> y=y_1+y_2
{y = y_1+y_2

Warning: function not tested for Kmat stored in sparse format

adjust_phase¶

Modify the phase y of a frequency response to remove discontinuities.

build_SS_poly¶

Builds a discrete-time state-space representation of a polynomial system whose frequency response has from:

Ypoly[oo,ii](k) = -A2[oo,ii] D2(k) - A1[oo,ii] D1(k) - A0[oo,ii]

where C1,D2 are discrete-time models of first and second derivatives, ds is the time-step and the coefficient matrices are such that:

A{nn}=Acf[oo,ii,nn]

butter¶

build MIMO butterworth filter of order ord and cut-off freq over Nyquist freq ratio Wn. The filter will have N input and N output and N*ord states.

Note: the state-space form of the digital filter does not depend on the sampling time, but only on the Wn ratio. As a result, this function only returns the A,B,C,D matrices of the filter state-space form.

compare_ss¶

Assert matrices of state-space models are identical

couple¶

Couples 2 dlti systems ss01 and ss02 through the gains K12 and K21, where K12 transforms the output of ss02 into an input of ss01.

Other inputs: - out_sparse: if True, the output system is stored as sparse (not recommended)

disc2cont¶

Transform a discrete time system to a continuous time system using a bilinear (Tustin) transformation.

Given a discrete time system with time step $\Delta T$, the equivalent continuous time system is given by:

\[\begin{split}\bar{A} &= \omega_0(A-I)(I + A)^{-1} \\ \bar{B} &= \sqrt{2\omega_0}(I+A)^{-1}B \\ \bar{C} &= \sqrt{2\omega_0}C(I+A)^{-1} \\ \bar{D} &= D - C(I+A)^{-1}B\end{split}\]

where $\omega_0 = \frac{2}{\Delta T}$.

References

MIT OCW 6.245

param sys: SHARPy discrete-time state-space object.
type sys: libss.ss
returns: Converted continuous-time state-space object.
rtype: libss.ss

eigvals¶

Ordered eigenvalaues of a matrix.

param a: Matrix.
type a: np.ndarray
param dlti: If true, the eigenvalues are ordered by modulus, else by real part.
type dlti: bool
returns: ordered set of eigenvalues.
rtype: np.ndarray

freqresp¶

In-house frequency response function supporting dense/sparse types

Inputs: - SS: instance of ss class, or scipy.signal.StateSpace* - wv: frequency range - dlti: True if discrete-time system is considered.

Outputs: - Yfreq[outputs,inputs,len(wv)]: frequency response over wv

Warnings: - This function may not be very efficient for dense matrices (as A is not reduced to upper Hessenberg form), but can exploit sparsity in the state-space matrices.

get_freq_from_eigs¶

Compute natural freq corresponding to eigenvalues, eigs, of a continuous or discrete-time (dlti=True) systems.

Note: if dlti=True, the frequency is normalised by (1./dt), where dt is the DLTI time-step - i.e. the frequency in Hertz is obtained by multiplying fn by (1./dt).

join¶

Given a list of state-space models belonging to the ss class, creates a joined system whose output is the sum of the state-space outputs. If wv is not None, this is a list of weights, such that the output is:

y = sum( wv[ii] y_ii )

Ref: equation (4.22) of Benner, P., Gugercin, S. & Willcox, K., 2015. A Survey of Projection-Based Model Reduction Methods for Parametric Dynamical Systems. SIAM Review, 57(4), pp.483–531.

Warning: - system matrices must be numpy arrays - the function does not perform any check!

join2¶

Join two state-spaces or gain matrices such that, given:

\[\begin{split}\mathbf{u}_1 \longrightarrow &\mathbf{SS}_1 \longrightarrow \mathbf{y}_1 \\ \mathbf{u}_2 \longrightarrow &\mathbf{SS}_2 \longrightarrow \mathbf{y}_2\end{split}\]

we obtain:

\[\mathbf{u} \longrightarrow \mathbf{SS}_{TOT} \longrightarrow \mathbf{y}\]

with $\mathbf{u}=(\mathbf{u}_1,\mathbf{u}_2)^T$ and $\mathbf{y}=(\mathbf{y}_1,\mathbf{y}_2)^T$.

The output $\mathbf{SS}_{TOT}$ is either a gain matrix or a state-space system according to the input $\mathbf{SS}_1$ and $\mathbf{SS}_2$

param SS1: State space 1 or gain 1
type SS1: scsig.StateSpace or np.ndarray
param SS2: State space 2 or gain 2
type SS2: scsig.StateSpace or np.ndarray
returns: combined state space or gain matrix
rtype: scsig.StateSpace or np.ndarray

parallel¶

Returns the sum (or parallel connection of two systems). Given two state-space models with the same output, but different input:

u1 –> SS01 –> y u2 –> SS02 –> y

project¶

Given 2 transformation matrices, (WT,V) of shapes (Nk,self.states) and (self.states,Nk) respectively, this routine returns a projection of the state space ss_here according to:

Anew = WT A V Bnew = WT B Cnew = C V Dnew = D

The projected model has the same number of inputs/outputs as the original one, but Nk states.

random_ss¶

Define random system from number of states (Nx), inputs (Nu) and output (Ny).

scale_SS¶

Given a state-space system, scales the equations such that the original input and output, $u$ and $y$, are substituted by $u_{AD}=\frac{u}{u_{ref}}$ and $y_{AD}=\frac{y}{y_{ref}}$.

If the original system has form:

\[\begin{split}\mathbf{x}^{n+1} &= \mathbf{A\,x}^n + \mathbf{B\,u}^n \\ \mathbf{y}^{n} &= \mathbf{C\,x}^{n} + \mathbf{D\,u}^n\end{split}\]

the transformation is such that:

\[\begin{split}\mathbf{x}^{n+1} &= \mathbf{A\,x}^n + \mathbf{B}\,\frac{u_{ref}}{x_{ref}}\mathbf{u_{AD}}^n \\ \mathbf{y_{AD}}^{n+1} &= \frac{1}{y_{ref}}(\mathbf{C}\,x_{ref}\,\mathbf{x}^{n+1} + \mathbf{D}\,u_{ref}\,\mathbf{u_{AD}}^n)\end{split}\]

By default, the state-space model is manipulated by reference (byref=True)

param SSin: original state-space formulation
type SSin: scsig.dlti
param input_scal: input scaling factor $u_{ref}$. It can be a float or an array, in which case the each element of the input vector will be scaled by a different factor.
type input_scal: float or np.ndarray
param output_scal: output scaling factor $y_{ref}$. It can be a float or an array, in which case the each element of the output vector will be scaled by a different factor.
type output_scal: float or np.ndarray
param state_scal: state scaling factor $x_{ref}$. It can be a float or an array, in which case the each element of the state vector will be scaled by a different factor.
type state_scal: float or np.ndarray
param byref: state space manipulation order
type byref: bool
returns: scaled state space formulation
rtype: scsig.dlti

series¶

Connects two state-space blocks in series. If these are instances of DLTI state-space systems, they need to have the same type and time-step. If the input systems are sparse, they are converted to dense.

The connection is such that:

\[u \rightarrow \mathsf{SS01} \rightarrow \mathsf{SS02} \rightarrow y \Longrightarrow u \rightarrow \mathsf{SStot} \rightarrow y\]

param SS01: State Space 1 instance. Can be DLTI/CLTI, dense or sparse.
type SS01: libss.ss
param SS02: State Space 2 instance. Can be DLTI/CLTI, dense or sparse.
type SS02: libss.ss

Returns: libss.ss: Combined state space system in series in dense format.

simulate¶

Routine to simulate response to generic input. @warning: this routine is for testing and may lack of robustness. Use

scipy.signal instead.

ss¶

class sharpy.linear.src.libss.ss(A, B, C, D, dt=None)[source]¶

Wrap state-space models allocation into a single class and support both full and sparse matrices. The class emulates

scipy.signal.ltisys.StateSpaceContinuous scipy.signal.ltisys.StateSpaceDiscrete

but supports sparse matrices and other functionalities.

Methods: - get_mats: return matrices as tuple - check_types: check matrices types are supported - freqresp: calculate frequency response over range. - addGain: project inputs/outputs - scale: allows scaling a system

addGain(K, where)[source]¶

Projects input u or output y the state-space system through the gain matrix K. The input ‘where’ determines whether inputs or outputs are projected as:

where=’in’: inputs are projected such that:
u_new -> u=K*u_new -> SS -> y => u_new -> SSnew -> y

where=’out’: outputs are projected such that:
u -> SS -> y -> y_new=K*y => u -> SSnew -> ynew

Warning: this is not a wrapper of the addGain method in this module, as the state-space matrices are directly overwritten.

disc2cont()[source]¶

Transform a discrete time system to a continuous time system using a bilinear (Tustin) transformation.

Wrapper of disc2cont()

eigvals()[source]¶

Returns: Eigenvalues of the system
Return type: np.ndarray

freqresp(wv)[source]¶

Calculate frequency response over frequencies wv

Note: this wraps frequency response function.

property inputs¶: Number of inputs $m$ to the system.

max_eig()[source]¶: Returns most unstable eigenvalue

property outputs¶: Number of outputs $p$ of the system.

project(WT, V)[source]¶

Given 2 transformation matrices, (WT,V) of shapes (Nk,self.states) and (self.states,Nk) respectively, this routine projects the state space model states according to:

Anew = WT A V Bnew = WT B Cnew = C V Dnew = D

The projected model has the same number of inputs/outputs as the original one, but Nk states.

scale(input_scal=1.0, output_scal=1.0, state_scal=1.0)[source]¶

Given a state-space system, scales the equations such that the original state, input and output, (x, u and y), are substituted by

xad=x/state_scal uad=u/input_scal yad=y/output_scal

The entries input_scal/output_scal/state_scal can be:

floats: in this case all input/output are scaled by the same value
lists/arrays of length Nin/Nout: in this case each dof will be scaled

by a different factor

If the original system has form:

xnew=A*x+B*u y=C*x+D*u

the transformation is such that:

xnew=A*x+(B*uref/xref)*uad yad=1/yref( C*xref*x+D*uref*uad )

property states¶: Number of states $n$ of the system.

transfer_function_evaluation(s)[source]¶

Returns the transfer function of the system evaluated at $s\in\mathbb{C}$.

Parameters: s (complex) – Point in the complex plane at which to evaluate the transfer function.
Returns: Transfer function evaluated at $s$.
Return type: np.ndarray

truncate(N)[source]¶: Retains only the first N states.

ss_block¶

class sharpy.linear.src.libss.ss_block(A, B, C, D, S_states, S_inputs, S_outputs, dt=None)[source]¶

State-space model in block form. This class has the same purpose as “ss”, but the A, B, C, D are allocated in the form of nested lists. The format is similar to the one used in numpy.block but:

Block matrices can contain both dense and sparse matrices

Empty blocks are defined through None type

Methods: - remove_block: drop one of the blocks from the s-s model - addGain: project inputs/outputs - project: project state

addGain(K, where)[source]¶

Projects input u or output y the state-space system through the gain block matrix K. The input ‘where’ determines whether inputs or outputs are projected as:

where=’in’: inputs are projected such that:
u_new -> u=K*u_new -> SS -> y => u_new -> SSnew -> y

where=’out’: outputs are projected such that:
u -> SS -> y -> y_new=K*y => u -> SSnew -> ynew

Input: K must be a list of list of matrices. The size of K must be compatible with either B or C for block matrix product.

get_sizes(M)[source]¶: Get the size of each block in M.

project(WT, V, by_arrays=True, overwrite=False)[source]¶

Given 2 transformation matrices, (W,V) of shape (Nk,self.states), this routine projects the state space model states according to:

Anew = W^T A V Bnew = W^T B Cnew = C V Dnew = D

The projected model has the same number of inputs/outputs as the original one, but Nk states.

Inputs: - WT = W^T - V = V - by_arrays: if True, W, V are either numpy.array or sparse matrices. If

False, they are block matrices.

overwrite: if True, overwrites the A, B, C matrices

remove_block(where, index)[source]¶

Remove a block from either inputs or outputs.

Inputs: - where = {‘in’, ‘out’}: determined whether to remove inputs or outputs - index: index of block to remove

ss_to_scipy¶

Converts to a scipy.signal linear time invariant system

param ss: SHARPy state space object
type ss: libss.ss
returns: scipy.signal.dlti

sum_ss¶

Given 2 systems or gain matrices (or a combination of the two) having the same amount of input/output, the function returns a gain or state space model summing the two. Namely, given:

u -> SS1 -> y1 u -> SS2 -> y2

we obtain:: u -> SStot -> y1+y2 if negative=False

Linear aeroelastic model based on coupled GEBM + UVLM¶

Linear aeroelastic model based on coupled GEBM + UVLM S. Maraniello, Jul 2018

LinAeroEla¶

class sharpy.linear.src.lin_aeroelastic.LinAeroEla(data, custom_settings_linear=None, uvlm_block=False, chosen_ts=None)[source]¶

Future work:

settings are converted from string to type in __init__ method.
implement all settings of LinUVLM (e.g. support for sparse matrices)

When integrating in SHARPy:

define:
- self.setting_types
- self.setting_default
use settings.to_custom_types(self.in_dict, self.settings_types, self.settings_default) for conversion to type.

Parameters

data (sharpy.presharpy.PreSharpy) – main SHARPy data class
settings_linear (dict) – optional settings file if they are not included in the data structure

settings¶

solver settings for the linearised aeroelastic solution

Type: dict

lingebm¶

linearised geometrically exact beam model

Type: lingebm.FlexDynamic

num_dof_str¶

number of structural degrees of freedom

Type: int

num_dof_rig¶

number of rigid degrees of freedom

Type: int

num_dof_flex¶

number of flexible degrees of freedom (num_dof_flex+num_dof_rigid=num_dof_str)

Type: int

linuvl¶

linearised UVLM class

Type: linuvlm.Dynamic

tsaero¶

aerodynamic state timestep info

Type: sharpy.utils.datastructures.AeroTimeStepInfo

tsstr¶

structural state timestep info

Type: sharpy.utils.datastructures.StructTimeStepInfo

dt¶

time increment

Type: float

q¶

corresponding vector of displacements of dimensions [1, num_dof_str]

Type: np.array

dq¶

time derivative ($\dot{\mathbf{q}}$) of the corresponding vector of displacements with dimensions [1, num_dof_str]

Type: np.array

SS¶

state space formulation (discrete or continuous time), as selected by the user

Type: scipy.signal

assemble_ss(beam_num_modes=None, wake_prop_settings=None)[source]¶: Assemble State Space formulation

get_gebm2uvlm_gains()[source]¶

Provides:

the gain matrices required to connect the linearised GEBM and UVLM

inputs/outputs

the stiffening and damping factors to be added to the linearised

GEBM equations in order to account for non-zero aerodynamic loads at the linearisation point.

The function produces the gain matrices:

Kdisp: gains from GEBM to UVLM grid displacements

Kvel_disp: influence of GEBM dofs displacements to UVLM grid velocities.

Kvel_vel: influence of GEBM dofs displacements to UVLM grid displacements.

Kforces (UVLM->GEBM) dimensions are the transpose than the

Kdisp and Kvel* matrices. Hence, when allocation this term, ii and jj indices will unintuitively refer to columns and rows, respectively.

And the stiffening/damping terms accounting for non-zero aerodynamic forces at the linearisation point:

Kss: stiffness factor (flexible dof -> flexible dof) accounting

for non-zero forces at the linearisation point. - Csr: damping factor (rigid dof -> flexible dof) - Crs: damping factor (flexible dof -> rigid dof) - Crr: damping factor (rigid dof -> rigid dof)